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effective alternative therapeutic regimens for ALD are urgently required. Various natural products exhibit strong capacities to scavenge free radicals and exert anti-inflammatory effects. These products are thus prime candidates for the prevention and/or treatment of alcoholic liver disease [17]. Aronia melanocarpa, also known as black chokeberry, is reported to be rich in anthocyanins and procyanidins [18]. A. melanocarpa is native to Russia and the midwestern United States but has been planted throughout northeastern Asia in regions such as Liaoning and Jilin Provinces in China [19]. A. melanocarpa and its metabolites, particularly procyanidins, exert antioxidative properties [20]. The anti-inflammatory effects of a crude extract of A. melanocarpa calyx have been confirmed in a rat model of lipopolysaccharide-induced uveitis [21]. Furthermore, the anthocyanin flavonoids produced by A. melanocarpa exerted a significant inhibitory effect on pancreatic swelling in a model of acute experimental pancreatitis [22]. Our group previously reported the ameliorating effects of A. melanocarpa fruit (AM) on gout and hyperuricemia in mice and rats via the regulation of redox imbalance [23]. These studies demonstrate that AM acts as a powerful antioxidant and anti-inflammatory agent. In this study, we proposed that AM could alleviate the alcoholic liver damage caused by long-term alcohol intake by inhibiting oxidative stress and reducing inflammation. Our findings from a C57BL/6 mouse model of alcoholinduced chronic liver injury confirm that AM prevents alcohol-induced liver injury by suppressing oxidative stress via the Nrf2 signaling pathway. AM Preparation and Composition Analysis. Fresh AM was obtained from Jilin Beijia Limited Company (Jilin, China) and homogenized. 2.1.2. Amino Acid | 3,069,200 | 210945092 | 0 | 16 |
Analysis. AM samples were hydrolyzed at 110°C for 24 h and then transferred to colorimetric tubes. The supernatants were removed, dried, and dissolved in 0.02 mol/L HCL. Five hundred microliters of the mixture were reacted with 250 μL of 0.1 mol/L phenyl isothiocyanate acetonitrile and 250 μL of 1 mol/L triethylamine acetonitrile for 1 h. After mixing this solution with 2 mL of n-hexane, the separated lower layer was collected and passed through a 0.45 μm organic film. Amino acids were analyzed quantitatively using liquid chromatography (1260, Agilent, Santa Clara, CA, USA). 2.1.3. Fatty Acid Analysis. A 5% KOH-methanol solution was added to the ME powder, placed in a 60°C water bath for 30 min, and then mixed with 14% boron trifluoridemethanol solution at 85°C for 30 min. The samples were then mixed with n-hexane and the levels of fatty acids analyzed using a gas chromatograph (7890A, Agilent). 2.1.5. Anthocyanidin Analysis. AM was ultrasonically extracted for 30 min in a hydrolysis tube and then hydrolyzed in a boiling water bath for 1 h. The samples were passed through a 0.45 μm aqueous phase filter and analyzed using an HPLC system equipped with a UV detector (LC-20AD, Shimadzu, Japan) and a C18 column (4:6 mm × 250 mm × 5 μm, 880975-902, Agilent, USA). The mobile phases included a 1% formic acid-water solution and 1% formic acid acetonitrile solution. The absorbance was measured at a wavelength of 530 nm. 2 Oxidative Medicine and Cellular Longevity Mice were randomly separated into five groups (n = 10 /group) and | 3,069,201 | 210945092 | 0 | 16 |
were intragastrically administered 10 mL/kg normal saline (normal control and model mice) or 63 mg/kg of silymarin (Sil; Madaus AG, Cologne, Germany; positive control mice) [30], as well as 0.5 or 2.0 g/kg of AM at 4:00 pm once per day for 24 weeks. During the first week, all but the control mice received drinking water containing 5% alcohol in sterilized tap water [31]. Gradually, the alcohol content was increased from 5% (v/v) to 30% (v/v), with a 5% increase each week [32]. All mice except those in the normal control group then received 30% alcohol until week 18. The body weights were monitored throughout the experimental period ( Figure 1). Blood and Liver Sample Collection. After the final treatment, the mice were fasted overnight and blood samples were collected from the caudal vein. All mice were then anesthetized and sacrificed. The liver, kidney, heart, and spleen were weighed on an electronic balance and stored immediately at -80°C. Histological Evaluation. Tissues were fixed in a solution of 4% paraformaldehyde in 0.1 M phosphate buffer as described previously [33], dehydrated in gradient alcohol, and embedded in paraffin. Five-micrometer sections were prepared and stained with hematoxylin and eosin (H&E) for histological evaluation. Kidney tissues were incubated in 0.1% periodic acid for 10 min, washed under running double distilled (D.D.) water for 1 min, and immersed in Schiff's reagent for 15 min. After washing with D.D. water for 3 min, the samples were counterstained with Mayer's hematoxylin for 2 min. After another 3 min wash, the samples were dehydrated | 3,069,202 | 210945092 | 0 | 16 |
twice in 96% alcohol. Finally, the samples were cleared in xylene and mounted with Entellan Neutral gum. The frozen liver tissue sections were warmed to room temperature for 10 min, then put in D.D. water for cleaning and immersed in sixty percent of isopropyl alcohol solution. After washing with isopropyl alcohol solution, put in Oil red O staining solution, dye for 10-15 minutes, and immerse in sixty percent of isopropyl alcohol solution again for color separation. After immersing, wash tissue twice in D.D. water and with tap water for five to ten minutes during the interval. Finally, sealed with glycerin gelatin. Oxidative Medicine and Cellular Longevity The bands were quantitated using Image J analysis software, version 1.46 (National Institutes of Health, Bethesda, MD, USA). Statistical Analysis. Data are expressed as means ± standard errors of the means. The post hoc Dunn's multiple comparison test was performed using SPSS 16.0 software (IBM Corp., Armonk, NY, USA) and analyzed using a oneway analysis of variance (ANOVA). Significance was defined as a p value <0.05. Hepatoprotective Effects of AM. Compared with normal control mice, a 24-week course of alcohol consumption led to a 31.2% decrease in body weight. Meanwhile, both AM and Sil administration prevented this reduction in body weight (p < 0:05; Table S1). Compared with the model mice, AM strongly suppressed swelling of the liver (p < 0:05 ), kidney (p < 0:01), and heart (p < 0:01) but had no effect on the spleen (Table S1). Sil only reduced liver swelling in mice with chronic alcohol-induced | 3,069,203 | 210945092 | 0 | 16 |
damage (p < 0:05), but not the swelling of other organs (Table S1). The serum AST/ALT levels served as biochemical markers of liver injury [34]. Compared with the normal control mice, alcohol administration strongly increased values of AST/ALT in both the serum and liver (p < 0:05), while AM administration prevented these increases (p < 0:05; Figure 2(a)). Subsequently, the antisteatosis effects of AM were further analyzed in mice with chronic alcohol-induced damage. AM increased the HDL levels > 33:0% (p < 0.01) (Figure 2(d)) and reduced the LDL levels by 14.0% (p < 0.01 Figure 2(c)) in the serum of 6-month alcohol-exposed mice. Neither AM nor Sil had a significant effect on the serum TG level (Figure 2(b)). In the livers of alcohol-exposed mice, AM administration resulted in 18.5%, 8.3%, and 20.2% reductions in the levels of TG (p < 0:01; Figure 2(b)), LDL (p < 0:05; Figure 2(c)), and AACS (p < 0:01; Figure 2(e)), respectively, and a >26.3% increase in the level of HDL (p < 0:001; Figure 2(d)). Sil strongly prevented the increases in the levels of TG (p < 0:001; Figure 2(b)) and LDL (p < 0:001; Figure 2(c)) and elevated levels of HDL (p < 0:01) in the liver (Figure 2(d)). The Oil red O staining further confirmed the protection of AM on the liver indicated by the reduced numbers of lipid droplets in the liver of chronic alcohol-damaged mice after AM and Sil administration (Figure 2(e)). Both AM and Sil strongly protected the liver against chronic alcohol damage, suggesting | 3,069,204 | 210945092 | 0 | 16 |
that these reagents alleviated the formation of lipid droplets and punctate necrosis and reduced the infiltration of inflammatory factors ( Figure 3). In the kidney, excessive alcohol caused narrowing of the glomerular capsule and hypertrophy of glomerular cells, both of which were significantly alleviated by AM and Sil as determined in H&E-stained sections ( Figure 3). PAS staining further confirmed that Sil and AM prevented kidney inflammation in chronic alcohol-damaged mice, as indicated by a reduction in PAS-positive areas ( Figure 3). Furthermore, AM and Sil suppressed the inflammatory infiltration into the hearts of mice subjected to chronic alcohol consumption. Antioxidative Effects of AM. Oxidative stress, which is caused by free radicals produced in response to alcohol consumption, has been reported as an important factor in the progression of liver disease [39]. Coenzyme Q3, CAT, and SOD are effective free radical scavengers and antioxidants, while MGST3 can effectively isolate harmful lipophilic compounds. Compared with mice exposed only to alcohol, AM prevented increases in the levels of MDA (p < 0:05), ROS (p < 0:05), APCS (p < 0:01), and CYP8B1 (p < 0:01) and reductions in the levels of CAT (p < 0:01), SOD (p < 0:05), GSH-Px (p < 0:05), CoQ3 (p < 0:05), and MGST3 (p < 0:001) in the livers of mice with chronic alcohol damage (Table 3). Sil exhibited similar antioxidative effects but failed to influence the hepatic levels of CoQ3 (Table 3). Oxidative Medicine and Cellular Longevity Regulatory Effects of AM on Nrf2 Signaling. PI3K/Akt signaling plays a critical role in | 3,069,205 | 210945092 | 0 | 16 |
the regulation of lipid metabolism and hepatocyte apoptosis in the liver and is regulated via a feedback loop with the Nrf2 signaling pathway [40]. Both AM and Sil enhanced the phosphorylation levels of PI3K (p < 0:05), Akt (p < 0:05), and STAT3 (p < 0:01) in the liver of mice with 6-month alcohol consumption (Figure 4(a)). Nrf2 signaling is involved in liver inflammation induced by long-term alcohol intake [41]. Both AM and Sil prevented the overexpression of Keap1 and reversed the low expression levels of Nrf2 (p < 0:05) and its downstream effectors, including HO-1 (p < 0:01), HO-2 (p < 0:01), SOD-1 (p < 0:01), and SOD-2 (p < 0:01) in the livers of alcohol-injured mice (Figure 4(a)). The low expression levels of Nrf2 in both the cytoplasm and nucleus were noted in the liver of long-term alcohol intake mice (p < 0:001), which were all strongly enhanced by AM and Sil (p < 0:001; Figure 4(b)). Discussion The nutritive value of AM was indicated by an analysis of its contents, which include 18 types of fatty acids, 17 types of amino acids, 4 types of anthocyanidins (including delphinidin and cyanidin), and 14 types of minerals. Previous studies identified anthocyanins among the most effective antioxidants and free radical scavengers, with various pharmacological efficacies [42]. Additionally, anthocyanin pretreatment significantly inhibits the alcohol-induced depletion of hepatic GSH and SOD [43]. AM is composed of approximately 0.32% anthocyanins, which provide a basis for the hepatoprotective properties of this product against alcoholinduced oxidative and inflammatory liver injury. In | 3,069,206 | 210945092 | 0 | 16 |
a mouse model of chronic alcoholic liver injury, AM effectively restored the AST/ALT ratio and alleviated pathological damage to the organs, particularly the liver. These findings from our study confirm the hepatoprotective effects of AM during chronic alcohol intake. During moderate or severe hepatitis, AST is released consequent to hepatocyte mitochondrial damage, leading to an increased AST/ALT ratio. This phenomenon is a hallmark of early liver injury [34]. AACS, as a lipid metabolism factor, helps to synthesize fatty acids and cholesterol, highly related with fat metabolism dysfunction [44]. Fat metabolism dysfunction is considered to be the progression of alcoholic fatty liver, which can further induce oxidative stress and inflammatory response [6,45]. AM effectively modulates the levels of AACS and reduces the numbers of lipid droplets in the liver, suggesting its inhibition on fat metabolism disorders. During the process of long-term alcoholic liver damage, alcohol is oxidized predominantly to acetaldehyde by alcohol dehydrogenase in hepatocytes, and this process promotes the synthesis of fatty acids. Eventually, this process leads to fatty liver and fat metabolism disorders, which are considered important processes in ALD and are closely related to oxidative stress [46]. AM effectively modulates the levels of factors associated with lipid metabolism, suggesting an ability of this product to prevent fat metabolism disorders and protect the liver. Metabolites such as ROS and acetaldehyde, which are generated during long-term alcohol consumption, are responsible for the inflammatory response in the liver. The levels of fibrinogens (e.g., FGL1 and FGG) are increased in most patients with severe liver disease [47]. An | 3,069,207 | 210945092 | 0 | 16 |
increased level of IL-2 is associated with hepatic fibrosis in humans with ALD All date are presented as mean ± S:E:M: (n = 10). # p < 0:05 and ## p < 0:01 compared with the control group; * p < 0:05, * * p < 0:01, and * * * p < 0:001 compared with the alcohol-only treated model group. All date are presented as mean ± S:E:M: (n = 10). # p < 0:05, ## p < 0:01, and ### p < 0:001 compared with the control group; * p < 0:05, * * p < 0:01, and * * * p < 0:001 compared with the alcohol-only treated model group. 8 Oxidative Medicine and Cellular Longevity Oxidative Medicine and Cellular Longevity [48]. A causal link has been established between aberrant levels of IL-4 and IL-6 and alcohol-induced hepatitis, primary biliary cirrhosis, and chronic hepatitis in humans [49,50]. This study has successfully confirmed the anti-inflammatory effects of AM in mice with long-term alcohol damage. Oxidative stress is a well-known cause of alcohol-induced liver injury, particularly as the liver is poorly resistant to excessive ROS [51]. ROS are responsible for inducing the inflammatory infiltration of neutrophils and the apoptosis and eventual necrosis of hepatocytes [52]. ROS is also known to affect lipid peroxidation [53]. An increase in the hepatic level of MDA is consistent with dyslipidemia, and reflects the degree of lipid peroxidation and the extent of hepatocyte damage [54]. As effective antioxidant enzymes, O 2and H 2 O 2 could be converted to H | 3,069,208 | 210945092 | 0 | 16 |
2 O by SOD and CAT inside cells [55], which has been reported as the first line of defense against ROS overaccumulation [56]. Furthermore, MGST3 participates in the detoxification process and effectively isolates harmful lipophilic compounds such as alcohol [57], thus protecting cells by preventing the oxidative destruction of sulfhydryl proteins [58,59]. CoQ3, one of the most effective free radical scavengers, helps to suppress endogenous free fatty acids [60], which are present at 10-fold higher levels in ALD patients than in healthy people [61]. CYP8B1 is required for the synthesis of cholic acid [62] that stimulates mitochondria to produce excess ROS in liver [63]. In this study, AM was observed to exert antioxidative effects by regulating various anti-and prooxidative factors in mice with alcohol-induced chronic liver injury. The overaccumulation of ROS in response to alcohol metabolism is sufficient to promote cellular necrosis, which in turn activates inflammatory responses and amplifies inflammatory signaling processes [64]. The activation of T cells by ROS induces the release of inflammatory factors, including IL-2, IL-4, and IL-6 [65]. Our data suggest that the hepatoprotective effects of AM against chronic alcohol damage are related to its excellent antioxidative properties. As a key transcription factor regulating oxidative damage, Nrf2 is regulated by ROS accumulation during alcohol administration. ROS modifies the thiol group in the intervention region (IVR) of Keap1, causing a conformational change in Keap1, resulting in its dissociation with Nrf2 or reduced ubiquitination of Nrf2, which then enter the nucleus to play an antioxidative role via regulating its downstream antioxidant enzymes, including | 3,069,209 | 210945092 | 0 | 16 |
SOD, CAT, and HO-1 [66]. STAT3, a key mediator of anti-inflammatory cytokines, is involved in Nrf2-medieated cytoprotection, which can be regulated by HO-1 [67]. The phosphorylated STAT3 can trigger the activation of PI3K/Akt signaling further silencing the increased phosphorylation levels of NF-κB and suppressing its downstream proinflammatory gene programs [68]. According to the previous study, Nrf2 can be regulated by PI3K/Akt signaling [69]. The pretreatment of PI3K inhibitor, LY294002, significantly inhibits the activation of Nrf2 [69]. The antioxidative property of AM is related to its modulation on Nrf2 signaling, which is further regulated by the phosphorylation activities of PI3K/Akt. There are still limits in this study. Based on the present data, we failed to conclude the main active ingredients con-tained in AM that show the protection on the liver against long-term alcohol exposure. Although the acute toxicity test has been performed in the preliminary experiment, which suggested the safe usage of AM on mice, more experiments should be applied to confirm its safety. Furthermore, during this study, the spleen and heart were found to have different degrees of damage. These injuries were related to long-term alcohol intake, and AM had protective effects on these organs. However, there is no in-depth report in this topic; we will conduct follow-up research. Taken together, this study first confirmed that 6-week AM administration can alleviate liver damage in mice induced by 24-week alcohol feeding through inhibition of oxidative stress and reduction of inflammatory reactions. The possible mechanism of AM's liver protection may be related to its regulation on Nrf2 signaling | 3,069,210 | 210945092 | 0 | 16 |
via the activities of PI3K/Akt in C57BL/6 mice with alcohol-induced chronic liver injury. Data Availability All generated and analyzed data used to support the findings of this study are included within the article. Conflicts of Interest The authors have declared that there is no conflict of interest. | 3,069,211 | 210945092 | 0 | 16 |
Note on the binomial partial difference equation Some formulas for the “general solution” to the binomial partial difference equation cm,n = cm−1,n + cm−1,n−1, are known in the literature. However, it seems that there is no such a formula on the most natural domain connected to the equation, that is, on the set D = { (m, n) ∈ N0 : 0 ≤ n ≤ m } . By using a connection with the scalar linear first order difference equation we show that the equation on the domain D \ {(0, 0)}, can be solved in closed form by presenting a formula for the solution in terms of the “side” values ck,0, ck,k, k ∈N. Introduction As we know, the Newton binomial formula can be written in the following form where m is an arbitrary natural number.The numbers C m n , 0 ≤ n ≤ m, are called the binomial coefficients, and from (1.1) and the relation Email: [email protected] m ≥ 2, it follows that and for every m ≥ 2. From (1.2) and the obvious fact which follows from (1.1) with m = 1, we also obtain for every m ∈ N. If we "naturally" assume (based, for example, on the combinatorial meanings of the coefficients), that is, introduce by a definition, that −1 = 0, m ∈ N, and C 0 0 = 1, then we see that such a defined double sequence C m n satisfies the relation in (1.3), for all m, n ∈ N 0 such that 0 ≤ n | 3,069,212 | 8092991 | 0 | 16 |
≤ m.All above mentioned is (or should be) nowadays known to any high-school student.For a good source of some classical things connected to this and related topics, see, for example, the nice problem book [9].Some more advanced results can be found, for example, in monographs [12] and [14]. Looking at recurrent relation (1.3) it can be seen that it is nothing but a camouflaged partial difference equation, which could be traditionally written in the following form more acceptable to the experts on difference equations.This seems one of the first partial difference equations appearing in the literature (it appeared much before than the notion partial difference equation was coined), and one of the basic ones (see, for example, [6, p. 1]).Some basic material on partial difference equations, especially related to the methods for solving linear and some related partial difference equations, can be found, for example, in the classical sources [8,Chapter 12] and [10,Chapter 8].A plenty of classical, as well as recent results on various types of linear and nonlinear partial difference equations, can be found, for example, in the nice monograph [6]. On the other hand, there has been some renewed recent interest in difference equations and systems which can be solved in closed form (see, for example, [1-4, 7, 13, 15, 16, 18-36, 38-47]).For some basic and classical types of solvable difference equations, see, for example, [8,10,12] (see, also, [9,Chapter 10], as well as some parts of the book [14]).Our note [15], in which a method for solving the difference equation appearing in [7] was | 3,069,213 | 8092991 | 0 | 16 |
given, triggered the renewed interest.Namely, it turned out that there are numerous nonlinear difference equations and systems of interest which can be transformed into the known solvable ones by using some suitable changes of variables.Many closed form formulas for solutions of the equations and systems seems obtained by using some computer packages, so they need some theoretical explanations.Some explanations for closed form formulas for solutions of such types of equations and systems can be found, for example, in [18,24,40].One of the crucial points in many of the above mentioned papers is the fact that the changes of variables transform the equations into the nonhomogeneous linear first order difference equation, that is, into the equation where (a n ) n∈N and (b n ) n∈N are arbitrary sequences of real or complex numbers, and x 0 ∈ R or x 0 ∈ C, respectively.For example, suitable changes of variables transform some of the difference equations in [1,2,4,13,15,16,19,23,34,40,41,43] into equation (1.5) or into its special cases.It should be pointed out that in some cases such obtained equations and systems are not equivalent to the original ones.Such a situation appears for example, in the original source [15], as well as in several later papers [19,23,28,39,45].Actually, in this or that way, many equations and systems are related to equation (1.5), or to the corresponding difference inequality or linear system of difference equations [5,11,12,36,39]. There are several methods for solving equation (1.5).For instance, by multiplying the equality by ∏ n j=n−l+2 a j , and summing up such obtained equalities for | 3,069,214 | 8092991 | 0 | 16 |
1 ≤ l ≤ n, it follows that which is the general solution to the equation.For some other methods, see, for example, [12]. For the case of some systems of difference equations the corresponding changes of variables transform them into some solvable linear ones (see, for example, [4, 18, 20-22, 24-30, 36, 39, 40, 44, 45]).In some other cases, such as in papers [32], [33], [35] and [46], where producttype systems related to the equations and systems in [17] and [37] are considered, or in papers [31,34,38,42], the transformations and methods used therein are more complex, but as a final outcome some solvable linear difference equations of higher order or more complicated solvable linear systems are obtained.If we also note that many of the solvable higher-order difference equations appearing in these papers can be presented as operator products of some linear first-order ones, we see an exceptional importance of equation (1.5).Now recall another well-known fact regarding the binomial coefficients.Namely, there is a concrete formula for them.It is which is one of the most basic formulas not only in combinatorics, but in mathematics as whole. The fact that the double sequence C m n is a solution to equation (1.4) suggests that the equation could be "solvable" in closed form.This is, in a way, true.Namely, for the partial difference equation (1.4) it is possible to find its "general solution" (the notion is more obscure than the corresponding one for the scalar difference equation and we will give some additional comments on it below).Indeed, if the following two | 3,069,215 | 8092991 | 0 | 16 |
operators are defined as Eu m,n = u m+1,n and Fu m,n = u m,n+1 , (see [10, p. 239]) then equation (1.4) can be written in the form which is the "general" solution to equation (1.4), since c 0,n , n ∈ Z, is an arbitrary sequence (i.e.function) on Z.To find a concrete solution to equation (1.4), it is clear that the initial values should be given at the points (0, n) n∈Z on the y-axis.This means that the "general" solution to equation (1.4) corresponds to the right-half plane.This clearly shows that beside a partial difference equation, the types of domains involved considerably influence on the "general" solution to the equation. The particular solution to equation (1.4) given in (1.7) is obtained from the "general" one by choosing the boundary values c 0,k as follows c 0,0 = 1 and c 0,k = 0, k = 0. However, the initial values appearing on y-axis seem less natural for the binomial coefficients and for getting formula (1.7), due to the fact that the natural domain for them is the set 0 and C m m , m ∈ N 0 , are specified.So, it is a natural question whether there is a closed form formula for solutions to equation (1.4) which reconstructs its solutions on domain D by using the given "side" values c 0,k and c k,k , when k ∈ N. Our aim here is to present a closed form formula for solutions to equation (1.4) in domain D \ {(0, 0)}.The formula could | 3,069,216 | 8092991 | 0 | 16 |
be known, but we could not locate it in the literature.Beside this, the formula seems quite unknown to a wide audience, so deserves publication in a visible place.Another aim is to point out a strong connection of a partial difference equation and its solvability with the solvability of the linear difference equation of first order. Main result Here, we first give a motivation for the main formula obtained in this paper, and show its connection with a special case of equation (1.5).The method presented here is half-constructive.Namely, we will first solve equation (1.4) in some special cases and then based on the obtained formulas we will assume the form of the general solution to the equation and confirm it by induction.The main idea is to note that for a fixed m equation (1.4) is actually equation (1.5) with which is, among others, why we paid some attention to equation (1.5) in the introduction, as well as to its usefulness in various applications.Although, in this case the corresponding linear equation is much simpler than the one in (1.5) and its solution is essentially obtained by the telescoping method, we want to point out the connection since we highly expect that for some related partial difference equations we might arrive at a position to use more general form of equation (1.5).We also want to emphasize that, because of the specificity of the domain, we will use its natural division by the lines y = x + k, where x ∈ N 0 , and where k is a | 3,069,217 | 8092991 | 0 | 16 |
fixed natural number.Namely, we will find a formula for the solution on the line with coefficient k, and then by using the formula on the line we will find a formula for the solution on the line with coefficient k + 1.Since k is an arbitrary natural number, this decomposition will produce the solution on the whole domain D. To demonstrate the method, and to point out clearly connection of equation (1.4) with a scalar linear difference equation of first-order, we will first find the solution to equation (1.4) for the case when k = 1, 2, 3. First assume that k = 1, that is, m = n + 1.Then in this case equation (1.4) becomes Summing up the equations in (2.1) from 1 to n, or equivalently solving it by using the change of variables which transforms equation (2.1) into the following linear difference equation of first-order and using formula (1.6) with a n = 1 and b n = c n,n , we get If we put it into (1.4),we get Similarly to the previous case, we get for every n ∈ N 0 .Using (2.2) in (2.4), change the order of summation and by some calculation, we get for every n ∈ N 0 .If k = 3, then m = n + 3, and if we put it in (1.4), we get S. Stević Similar to the case k = 1, we get for every n ∈ N 0 .Using (2.5) in (2.7), changing the order of summation and by some | 3,069,218 | 8092991 | 0 | 16 |
calculation, we get where we have also used the well-known formula Hence, on the lines m = n + k, for k ∈ {1, 2, 3}, we have found the solution to equation (1.4) by constructing it.The procedure can be continued for k = 4 and other small values of k.However, since we need a closed form formula to solutions of equation (1.4) which holds for every m ∈ N, we need a formula for c n+k,n which holds for every k ∈ N. Formulas (2.2), (2.5) and (2.8) suggest that the following formula holds for every n ∈ N 0 and k ∈ N. We apply the method of induction.According to the previous consideration equality (2.9) holds for k = 1 since (2.2) can be written in the form Assume that (2.9) hold for some k ∈ N. If we put m = n + k + 1 into (1.4),we get Using the hypothesis (2.9) in (2.11), and by some simple calculations, we get for every n ∈ N 0 .By using recurrent relation (1.3), we have for every 1 ≤ r ≤ k, and for every 1 ≤ i ≤ n.Using (2.13) and (2.14) in (2.12), it follows that from which along with the method of induction it follows that formula (2.9) holds for every n ∈ N 0 and k ∈ N. Due to the above considerations, we are now in a position to formulate and prove the main result in this note. is given by .16) Proof.If we put k = m | 3,069,219 | 8092991 | 0 | 16 |
− n in formula (2.9) (note that m > n so k ∈ N), use the "side" conditions given in (2.15), and by some simple calculations, we obtain formula (2.16). Remark 2.2.Note that Theorem 2.1 actually says that the general solution to partial difference equation (1.4) on domain D \ {(0, 0)} is given by the formula which is the closed form formula that we wanted to obtain (the formula which reconstructs the solutions to equation (1.4) by given "side" values c k,0 , c k,k , k ∈ N). S. Stević Remark 2.3.Note that formula (2.17) does not contain value c 0,0 , which is why instead of domain D we consider domain D \ {(0, 0)}. is well-known and can be found in many books on combinatorics or problem books on elementary mathematics, in this or some equivalent forms (see, e.g., [9] or [12]).We have added the proof for its simplicity, benefit of the reader and for the completeness of the note. Theorem 2 . 1 . If (u k ) k∈N , (v k ) k∈N , are given sequences of real numbers.Then the solution to partial difference equation (1.4) on domain D \ {(0, 0)}, with the boundary value conditions given by c k,0 = u k and c Corollary 2 . 4 .From ( 2 . The solution to partial difference equation (1.4) on domain D \ {(0, 0)}, with the boundary value conditions given by c k,0 = 1 and c k,k = 1, k ∈ If we put the | 3,069,220 | 8092991 | 0 | 16 |
conditions in (2.18) in formula (2.16), we get c m,n = 22) and since C m m−n = C m n , formula (2.19) follows, as desired. | 3,069,221 | 8092991 | 0 | 16 |
Floquet-Mode Traveling-Wave Parametric Amplifiers Simultaneous ideal quantum measurements of multiple single-photon-level signals would advance applications in quantum information processing, metrology, and astronomy, but require the first amplifier to be simultaneously broadband, quantum limited, and directional. However, conventional traveling-wave parametric amplifiers support broadband amplification at the cost of increased added noise and are not genuinely directional due to non-negligible nonlinear backward wave generation. In this work, we introduce a new class of amplifiers which encode the information in the Floquet modes of the system. Such Floquet mode amplifiers prevent information leakage and overcome the trade-off between quantum efficiency (QE) and bandwidth. Crucially, Floquet mode amplifiers strongly suppress the nonlinear forward-backward wave coupling and are therefore genuinely directional and readily integrable with qubits, clearing another major obstacle towards broadband ideal quantum measurements. Furthermore, Floquet mode amplifiers are insensitive to out-of-band impedance mismatch, which otherwise may lead to gain ripples, parametric oscillations, and instability in conventional traveling-wave parametric amplifiers. Finally, we show that a Floquet mode Josephson traveling-wave parametric amplifier implementation can simultaneously achieve $>\!20\,$dB gain and a QE of $\eta/\eta_{\mathrm{ideal}}\!>99.9\%$ of the quantum limit over more than an octave of bandwidth. The proposed Floquet scheme is also widely applicable to other platforms, such as kinetic inductance traveling-wave amplifiers and optical parametric amplifiers. Faithful amplification and detection of weak signals are of central importance to various research areas in fundamental and applied sciences, ranging from the study of celestial objects in radio astronomy and metrology [1,2], dark-matter detection in cosmology [3][4][5], and exploration of novel light-matter interactions in atomic | 3,069,222 | 248060231 | 0 | 16 |
physics [6] to superconducting [7][8][9] and semiconductor spin [10,11] qubit readout in quantum information processing. In circuit quantum electrodynamics (cQED), near-quantum-limited amplifiers enable fast high-fidelity readout and have helped achieve numerous scientific advances, such as the observation [12] and reversal [13] of quantum jumps, the "break-even" point in quantum error correction [14][15][16], and quantum supremacy or quantum advantage [17]. Josephson traveling-wave parametric amplifiers (JTWPAs) [18][19][20] with several gigahertz of bandwidth and a dynamic range of P 1dB approxmately −100 dBm are widely used as preamplifiers in microwave quantum experiments. While near-ideal intrinsic quantum efficiency (QE) has been achieved with Josephson parametric amplifiers (JPAs) [21][22][23][24][25], the behavior of which is well understood [26,27], the best reported intrinsic QE of JTWPAs remains at least 20% below that of * Correspondence email address: [email protected] an ideal phase-preserving amplifier, despite several independent implementations [28,29]. Although such reductions in intrinsic QE are commonly attributed almost entirely to dielectric losses, detailed noise characterization suggests that an unknown noise mechanism is dominant [19]. Such a yet to be identified noise source will limit the best achievable readout fidelity and speed, and eventually hinders the realization of broadband ideal quantum measurements which are critical to continuous quantum error correction [30,31], quantum feedback control [32][33][34][35], and ultrasensitive parameter estimation [36]. Consequently, an outstanding, yet unanswered, question is: can a wideband parametric amplifier achieve near-ideal quantum efficiency? In this work, we quantitatively identify the sidebands as the dominant noise mechanism in existing TWPAs using the multimode, quantum input-output theory framework presented here, which also models propagation | 3,069,223 | 248060231 | 0 | 16 |
loss quantum mechanically. We then offer an affirmative answer to the previous question by introducing Floquet mode amplifiers, a new class of broadband amplifiers that encode information in Floquet modes and effectively eliminate coherent information leakage. A Floquet mode JTWPA can achieve > 20 dB gain and a QE of η/η ideal > 99.9% of the quantum limit over an instantaneous 3 dB bandwidth of approximately 6.5 GHz. Importantly, Floquet mode TWPAs strongly suppress the nonlinear forward-backward-mode couplings, which dominate the signal reflection in conventional homogeneous critical current TWPAs that are well impedance matched. Floquet mode TWPAs are thus genuinely directional and can minimize signal reflection to < −25 dB over the full amplifying bandwidth, making them integrable with qubits without commercial isolators and potentially enabling near-perfect full-chain measurement efficiency. Additionally, Floquet mode TWPAs also offer the practical advantages of convenient interfacing using bare frequency modes and insensitivity to out-of-band impedance environment, which strongly suppresses gain ripples. We predict a QE of η/η ideal > 99.9% to be realistically achievable using a fabrication process with a dielectric loss tangent tan δ ∼ 10 −6 , typical of qubit fabrication. Although here we illustrate the new Floquet mode amplifier using a microwave JTWPA design, it is worth noting that it can be readily applied to any traveling-wavestyle parametric amplifiers such as the kinetic inductance traveling-wave amplifiers (KITs) [37][38][39]. We anticipate that Floquet mode TWPAs will help advance various information-critical applications in metrology and quantum information processing,enabling the longstanding goal of broadband high-sensitivity dark-matter searches [5] and paving | 3,069,224 | 248060231 | 0 | 16 |
the way for scalable fault-tolerant quantum computing by enabling fast, multiplexed qubit readout below the surface code error threshold [40]. I. MULTIMODE DYNAMICS The uncertainty principle of quantum mechanics requires any linear phase-preserving amplifier to add at least approximately a half quantum of noise referred to the input at high gain [41][42][43]. For an ideal two-mode parametric amplifier, the idler mode, described by the creation operatorâ † (ω i ), acts as the coherent "reservoir" that injects the minimal added noise to the signalâ(ω s ) to preserve the bosonic commutator relations at the output. At a power gain G ≥ 1 in linear units, the ideal quantum efficiency is commonly defined as [27] whereâ in(out) (ω s ) is the input (output) annihilation operator of the signal, and |∆â| 2 ≡ {â,â † } / 2 − â † â is the mean-square fluctuation of operatorâ [42]. It is worth emphasizing that η ideal corresponds to the standard quantum limit even though it is only approximately 50% at high gain G. No information is lost in the process [43] because the idler is not correlated with any other unmeasured degrees of freedom. Such ideal two-mode amplifiers, albeit halving the measurement strength, can preserve information perfectly as illustrated in Fig. 1(a). Practical TWPAs, however, are intrinsically multimode because their large bandwidth allows for a spectrum of sidebands to propagate simultaneously. Sidebands have been a longstanding problem for all types of wideband TWPAs [44,45]. The same pump tone providing the large signal gain necessary to suppress the | 3,069,225 | 248060231 | 0 | 16 |
excess noise of downstream amplifiers also unavoidably induces the sideband couplings, leading to non-negligible noise and information leakage as illustrated in Fig. 1(b). Although such sideband couplings are also present in our proposed Floquet mode amplifiers, we show later that Floquet mode amplifiers are nevertheless still able to circumvent this issue and reach the quantum limit over their broad working bandwidth. This can be intuitively understood either as the Floquet mode amplifiers coherently recovering the leaked information back into the signal and idler mode in the bare frequency picture [ Fig. 1(c)] or as encoding information instead in the collective Floquet modes of the driven system and adiabatically mode matching to the bare frequency modes at input and output for convenient interfacing in the Floquet-mode picture. The various two-pump-photon parametric interactions responsible for the coherent information leakage in a typical degenerate four-wave mixing (4WM) TWPA can be conveniently visualized using the two-sided frequency spectrum in Fig. 2(b). In this representation, the idler mode has a defined negative frequency of ω −1 = ω s − 2ω p = −ω i in recognition of the relationâ(−ω i ) =â † (ω i ) [46]. Similarly, the frequencies of the gray-colored sideband modesâ n =â(ω n ) are specified by ω n = ω s + 2nω p , in which the integer mode index n can be either negative or positive. Following this convention, 4WM processes couple only adjacent modes that are spaced by 2ω p , whereas six-wave-mixing (6WM) processes can instead couple modes separated by | 3,069,226 | 248060231 | 0 | 16 |
up to two spacings and so forth. Frequency conversion (FC) and parametric amplification (PA) processes can be distinguished by whether the frequencies of their two interacting modes possess the same or opposite signs, respectively. Analogously to The circuit diagram of a generic resonantly phase-matched JTWPA terminated by 50 Ω linear transmission line ports at x = 0 and x = L, where L is the device length in unit cells (of length a). The unit cell at x consists of a pair of ground capacitors Cg(x)/2 and a Josephson junction with junction energy EJ (x) = φ0I0(x) and capacitance CJ (x), in which φ0 = Φ0/(2π) is the reduced flux quantum, and I0(x) is the junction critical current. Phase-matching resonators with inductance Lr, capacitance Cr, and coupling capacitance Cc(x) are inserted periodically for phase matching. The dielectric loss is modeled by a series of lossless transmission line ports in light gray, the scattering parameters of which are determined by the loss rate γ(ω). f ± n (x) are the uncorrelated noise operators introduced to the system by distributed loss. (b) A two-sided frequency spectrum representation of a typical degenerate four-wave mixing TWPA. The signal and idler frequencies are denoted as ω0 and ω−1, respectively. The solid and dashed arrows indicate the parametric amplification and frequency conversion processes, respectively. The example frequency scale corresponds to a 6-GHz signal and a 8-GHz pump. (c) A Sankey-like illustration of the output operator or noise decomposition of a multimode TWPA constructed from the scattering matrices S0 and Sn(x). The relative | 3,069,227 | 248060231 | 0 | 16 |
width of each "water flow" from source node i to target node j represents the operator or noise composition of the input modeâi in the output modeâj. a qubit, information can coherently leak out from the "computational subspace" of the signal and the idler in the yellow-shaded region of Fig. 2(b) into the sidebands, in addition to incoherent losses from radiation or dissipation. The effects of sidebands on phase matching and hence gain dynamics have been studied in KITs and JTWPAs [47][48][49] but the connection between sidebands and the noise performance has not been recognized, potentially due to the lack of a systematic rigorous quantum framework. To quantify the effects of both sideband leakage and propagation loss on the QE of TWPAs, we develop a multimode quantum input-output theory framework, as illustrated in Fig. 2(a). The circuit design exemplified in the figure is similar to, but more general than, that of a typical resonantly phase-matched JTWPA [18,19], as it allows the device to be inhomogeneous and the circuit parameters to have a spatial dependence. The propagation loss is modeled quantum mechanically with a series of distributed, lossless, and semi-infinite transmission line ports. Dissipation and their associated fluctuations can then be cast as the coherent scattering into and from these transmission lines, the frequency-dependent scattering parameters of which are set by the loss rate γ(ω) [50][51][52]. Furthermore, we extend the beam-splitter model such that our model can now work for any generic second-order equation of motion and properly account for the loss and interactions among the forward | 3,069,228 | 248060231 | 0 | 16 |
and backward modes without taking the slowly varying envelope approximation (for details, see Appendix B 2). In addition, our model can correctly account for impedance mismatch at the boundaries, insertion loss, and nonlinear processes from arbitrary orders of pump nonlinearities (4WM, 6WM, · · · ). Under the stiff-pump approximation, the multimode system can be linearized around a strong, classic pump with a dimensionless amplitude I pn (x) = I p (x)/I 0 (x), where I p (x) and I 0 (x) are the pump current and the junction critical current at x, respectively. In the continuum limit (|k s,i |a 1), the quantum spatial dynamic equations in the frequency domain can be derived from the Heisenberg equations and written in block matrix form as follows: in which A ± (x) = [· · · ,â ± n (x), · · · ] T are the forwardand backward-propagating operator vectors, F ± (x) are, similarly, the forward and backward noise operator vectors, Γ(x) = diag [· · · , γ(ω n ), · · · ] is the diagonal loss-rate matrix, and 0 m is the m × m zero matrix, with m be-ing the number of frequency modes considered. The field ladder operatorsâ ± n (x) satisfy the commutation relations [46] â d1 n1 (x),â d2 † n2 (x ) = sgn(ω n1 )δ n1,n2 δ d1,d2 δ(x − x ), (3) in which the superscripts d 1 , d 2 ∈ {+, −} denote the forward-(+) or backward-(-) propagating modes and sgn(ω) is the | 3,069,229 | 248060231 | 0 | 16 |
sign function. The noise operatorsf ± n (x) follow a commutation relation that is similar to Eq. (3). The multimode coupling matrix K(x) in the first term of Eq. (2) is positive definite and captures the various PA, FC, and forward-backward coupling processes (see Appendix B 2), whereas the second and third terms describe the dissipation and the associated fluctuation from material loss respectively. By solving Eq. (2) and applying the proper boundary conditions at x = 0 and x = L, we can relate the input and output bosonic modes A in and A out in the transmission lines ports terminating the TWPA by (see Appendix C) where L is the device length in unit cells and S 0 and S n (x) are the 2m × 2m multimode and noise scattering matrices that capture the effects of sideband couplings and dissipation on the output signal, respectively. Together, they are sufficient to calculate the full quantum statistics of the output modes. The quantum efficiency η of a general multimode TWPA with dissipation can thus be calculated from S 0 and S n (x) as where A k denotes the k-th operator of the vector A ∈ { A in , F(x)}, s is the index of the forward signal modeâ + 0 , and |S n,sk | 2 is the signal power gain G. Note that Eq. 2(c) pictorializes the dynamics of a multimode TWPA from the perspective of added noise using information from the scattering matrices S 0 and S n (x). A "water | 3,069,230 | 248060231 | 0 | 16 |
flow" from source mode i to target mode j can be equally interpreted as the operator or noise composition of input mode i in the output mode j. The noise of component i in output j is proportional to the width of the i → j path. The quantum efficiency η therefore corresponds to the weight (width) of the input signal noise in the total output signal noise (width). In the case of an ideal two-mode parametric amplifier, the output signal would have all of its in-flow (noise) coming from only the input signal and idler, signifying quantum-limited noise performance. In contrast, as is shown for a practical TWPA, an additional, non-negligible portion of the output signal noise instead comes from the input sideband modes and the reservoir of fluctuation operators, degrading the QE. II. THE FLOQUET-BASIS PICTURE We now numerically simulate and visualize the spatial dynamics inside a JTWPA, the circuit parameters of which are similar to those in Ref. [19], except that here we assume tan δ = 0 here (see Table I in Appendix A). In Figure 3(a), a unit of forward signalâ + 0 is injected from x = 0 with I pn ≡ I p /I 0 = 0.52, which is chosen to produce approximately 20 dB gain in L = 2037 cells. Although still being amplified, the forward signal and idler are continuously being converted to the sidebandŝ a + 1 andâ + −2 , leading to the sideband amplification and information leakage. Figure 3(b) plots the noise decomposition of | 3,069,231 | 248060231 | 0 | 16 |
the amplified output signalâ + 0 (x = L + ) in the 50 Ω output port as a function of I pn . The device length is fixed at L = 2037, such that the signal gain increases nearly monotonically with I pn before the onset of parametric oscillations. From Eq. (5), the QE η can be usefully interpreted as the ratio of the amount of noise from the original signalâ + 0 to the total output signal noise. QE therefore maps exactly to the signal contribution (in solid blue) in Fig. 3(b), which decreases with the I pn due to an elevated sideband contribution. This quantitatively accounts for the unknown QE reduction in Ref. [19] and provides numerical evidence that the sidebands are indeed a significant noise source in JTWPAs. Floquet theory [53] provides invaluable insights into the noise performance of TWPAs. Floquet modes are a set of solutions for a periodically driven system that forms a complete orthonormal basis. Each Floquet mode can be expressed asQ α (x) = e rαxα (x), whereα(x) is spatially periodic and r α is the complex Floquet characteristic exponent of Floquet mode α. For a homogeneous lossless TWPA described by Eq. (2), the solution can be expressed in the form of , in which the transfer matrix Π(x) can be written in the form of Π(x) = P(x)exp(xQ) according to the Floquet theorem and P(x) has the same periodicity as K(x). We can thus transform the frequency-basis vector A(x) into the Floquet-mode basis Q(x) via (for details, | 3,069,232 | 248060231 | 0 | 16 |
see Appendix E) where V is the orthonormal basis of matrix Q such that Q = VΛV −1 with Λ = diag(· · · , r α , · · · ) containing the Floquet exponents. Figure 3(c) shows the exact same spatial evolution as in Fig. 3(a) but in the Floquet-mode basis. A unit of forward input signalâ + 0 (0) is injected and projected into a collection of Floquet modes, each of which then propagates separately with a distinct complex propagation constant r α . Notably, there is only one amplifying and one deamplifying Floquet modeQ a andQ d , which can be understood as the antisqueezing and squeezing quadratures of the signal-idler-like mode as shown in Fig. 3(d). All the other sidebandlike Floquet modes, colored in gray, remain constant in space (for details, see Appendix E). This suggests an alternative view to the sideband-induced excess noise and coherent information leakage: they result from the mode mismatch between the bare frequency modes of the input and output ports and the collective Floquet modes of a driven TWPA. At G 1, the amplifying Floquet modeQ a dominates the gain and noise performance of a TWPA. Figure 3(d) plots the frequency-mode decomposition ofQ a as a function of I pn . The Hopf bifurcation point of the signal and idler components at I pn ≈ 0.02 marks the transition of the system from the region of stability to instability (amplification), the exact position of which is dependent upon the phase mismatch. The mixing of signal and | 3,069,233 | 248060231 | 0 | 16 |
idlers and the bifurcation of complex-conjugate pairs r a and r d [ Fig. 3(e)] implies the standard quantum limit for phase-preserving amplifiers: signal-idler mixtures of different relative phases are split into the amplifying and deamplifying Floquet modes and thus cannot be all noiselessly amplified at the same time. The gain coefficient g a = Re{r a } ofQ a [ Fig. 3(e)] increases monotonically with I pn andQ a mixes in more sideband modes; in particular,â 1 andâ −2 . The remarkable resemblance between Fig. 3(b) and Fig. 3(d) in the region of large signal gain shows the usefulness of the Floquet basis in understanding the TWPA noise performance. An increased sideband weight inQ a means that a larger por-tion of the sideband vacuum fluctuations incident upon a TWPA would be projected into the amplifying Floquet mode and would then subsequently generate more noise power at the signal and idler frequencies. The dependence of theQ a mode mixture on I pn also sheds light on the experimental observation that the peak signal-tonoise ratio (SNR) improvement often does not coincide with the largest signal gain as a function of pump power [19]. Figure 3(e) plots in three dimensions the complex Floquet characteristic exponents r α as a function of I pn (x axis). All but the amplifying and deamplifying Floquet modes stay within the plane of g α = Re{r α } = 0 throughout and are stable, whereas the gain coefficient magnitudes ofQ a andQ d increase with I pn as expected. It is important | 3,069,234 | 248060231 | 0 | 16 |
to point out that r a and r d , the complex eigenvalues ofQ a andQ d respectively, do not ever intersect with those of any other Floquet modes at all values of I pn , as made clear in the inset of Fig. 3(e). The existence of a gap between r a (r d ) and the rest of the spectrum will be crucial to the Floquet mode amplifiers introduced in Section III. III. THE FLOQUET MODE AMPLIFIERS We now introduce the Floquet mode amplifiers, which can effectively eliminate the aforementioned sidebandinduced noise and approach the standard quantum limit simultaneously over the broad operating bandwidth. As alluded to earlier in Section I and depicted in Fig. 4(a), the principal idea behind Floquet mode amplifiers is that they are effective two-"Floquet mode" amplifiers, as information is exclusively encoded in the two instantaneous collective amplifying and deamplifying Floquet modes, Q a (x) andQ d (x), of the driven system. Furthermore, we adiabatically transform the instantaneous Floquet modes inside the amplifier with a spatially varying dimensionless drive-amplitude profile I pn as illustrated in Fig. 4(b). This allows us to mode match the information-carrying Floquet modes to the input and output bare single frequency modesâ + 0 for convenient interfacing, as in the case of existing TWPAs. I pn (x) are ideally set to near zero at the boundaries to perfectly mode match the signal and idler modesâ + 0 andâ + −1 in the linear input and outputs to the instantaneousQ a (x) andQ d (x). Near the | 3,069,235 | 248060231 | 0 | 16 |
center of the amplifier, I pn (x) is adiabatically ramped up to significantly amplify the signal within a reasonable device length. Note that the increased sideband mixture inQ a (x) andQ d (x) in the middle does not contribute to additional noise in this scheme, because they will eventually be adiabatically transformed back toâ + 0 andâ + −1 as I pn (x) ramps down to near zero again in the end. As a result, from the view outside of the device, the signal and idler modes are effectively decoupled from the various sidebands and Floquet mode amplifiers can therefore approach quantum-limited noise performance. In practice, we can tailor the desired spatial profile of I pn (x) from a constant input pump current I p by instead varying the junction critical current I 0 (x) and ground capacitance C g (x) as shown in Fig. 4(b). Here, we consider the case in which the spatially varying I 0 (x) is achieved with varying junction areas but the same plasma frequency using a single fabrication step. We also vary the coupling capacitance C c (x) such that the cou- , which centers at µ = L/2 = 1000 and has a full width at half maximum (FWHM) of 2 √ 2 ln 2σ = 0.62L. This practical choice of FWHM leads to a small but nonzero I pn (x) ≈ 0.1 at the boundaries, but this still results in nearly ideal quantum efficiency, as it is close to the bifurcation point (I pn ≈ 0.05). Furthermore, here | 3,069,236 | 248060231 | 0 | 16 |
the minimum and maximum junction currents required to achieve an overall dynamic range of approximately −100 dBm are around 3.5 µA and 21.2 µA, both of which can be readily fabricated with Lecocq-style junctions [54] or in a niobium-trilayer process [55], demonstrating the practicality and robustness of our scheme. Figures 4(g) and 4(h) show, respectively, the internal field profiles of the homogeneous and Floquet scheme in the frequency basis when a forward input signalâ + 0 is injected at x = 0 − . While the signal and idlers are amplified by approximately 25 dB in both schemes, the Floquet scheme efficiently suppresses the sidebands and the backward modes. Figures 4(i) and 4(j) show the system response of both schemes when, instead, only the sideband vacuum fluctuationâ + 1 (0 + ) is injected. Whereas the conventional homogeneous design generates a significant amount of added noise at the signal and idler frequencies, the Floquet scheme minimizes the coupling fromâ + 1 toQ a and thus suppresses the sideband-induced noise by several orders of magnitude, thereby attaining near-ideal quantum-limited noise performance. To clearly distinguish the noise performance of different near-quantum-limited amplifiers, we define the amplifier quantum inefficiency as ... ... which signifies the relative difference in the resulting output SNR between a realistic and an ideal phasepreserving amplifier at the same power gain. An ideal preserving amplifier will therefore, by definition, have an inefficiency ofη = 0, denoting the standard quantum limit. In Figs. 4(c) and 4(d), we plot, respectively, the simulated gain and quantum inefficiency spectrum | 3,069,237 | 248060231 | 0 | 16 |
of the proposed Floquet scheme and the conventional homogeneous design. We also include the simulated performance of the experimental device from Ref. [19] at a similar gain level for comparison. Our multimode quantum model predicts a quantum inefficiencyη = 0.13 or η/η ideal = 87% for the experimental device assuming no dielectric loss, which is in good agreement with the experimentally extracted value of η/η ideal = 85% in Ref. [19]. This suggests that our multimode quantum model is able to accurately predict and identify the previously unknown experimentally measured noise mechanism as the sideband-induced noise. For both conventional homogeneous schemes, the quantum inefficiency is still on the order of 10 −1 away from the standard quantum limit due to the additional sideband-induced noise, although a design with a lower cutoff frequency of 65 GHz (blue curves in Figs. 4(c) and 4(d)] shows a slight improvement. Notably, sideband effects also manifest themselves in the vis-ible oscillations on the quantum inefficiency or noise spectrum of the conventional homogeneous schemes. Such oscillations in the amplifier added noise have been observed in experiments [29,39]. In contrast, the Floquet mode TWPA is able to both produce high gain and attain near-ideal QE over a large bandwidth of 6.5 GHz (after excluding the band gap due to the phase-matching resonators), well exceeding an octave in 2000 unit cells. The vanishingly small quantum inefficiency of the adiabatic Floquet design is a direct consequence of the effective decoupling of the signal and idler from the sidebands. The quantum inefficiencyη of the Floquet mode | 3,069,238 | 248060231 | 0 | 16 |
TWPA is shown in Fig. 4(h) to be smaller than 10 −3 over the full amplifying bandwidth, which is orders of magnitude closer to the quantum limit and can be practically realized. We note that the broadband signal gain of the Floquet mode TWPA can be further increased if desired, either by driving at a slightly larger I pn in situ with a minor decrease in quantum efficiency or with a slightly longer device such that the quantum efficiency remains similar (for details, see Appendix E 3). For instance, ≥ 30 dB gain can be achieved by driving the discussed Floquet TWPA design at I pn = 0.635 (6% increase) or with a longer device of 2100 cells (a 5% increase). In addition, the upper limit of the dynamic range of TWPA is dominated by pump depletion. The power dependence of the signal gain has the approximate form G(P s ) = G 0 /(1 + 2G 0 P s /P p ) [18], where G 0 is the small signal gain and P s (P p ) is the signal(pump) power. Because the cutoff frequency of a Floquet mode TWPA is determined by its smallest critical current junctions in the middle, the critical current of all Floquet mode TWPA junctions are no less than those of the corresponding conventional TWPA. Consequently, with the same driving strength I pn (x) and cutoff frequency f c , the pump power P p and thus the dynamic range of the Floquet TWPA are both larger or equal to | 3,069,239 | 248060231 | 0 | 16 |
those of its conventional TWPA counterpart. The 1-dB gain compression power of the presented Floquet TWPA design is estimated to be P 1dB ∼ −100.4 dBm, on par with those of conventional TWPAs reported in the literature [56]. IV. DIELECTRIC LOSS, DIRECTIONALITY, AND ON-CHIP INTEGRATION We now discuss the nonideality of finite dielectric loss with Fig. 5(a), in which the quantum inefficiencyη at 6 GHz of the three designs in Fig. 4 is computed as a function of the loss tangent tan δ with all other conditions fixed. Here, we neglect pump attenuation due to the dielectric loss, as it can be compensated by adjusting the circuit parameters accordingly in the adiabatic Floquet scheme. The left and right vertical gray dashed lines correspond to tan δ = 3.4×10 −3 of the SiO x capacitors in Refs. [19,55] and tan δ = 10 −6 of a typical qubit fabrication process [57,58], respectively. With SiO x capacitors, the calculated quantum inefficiency of the homogeneous TWPA in Ref. [19] increases toη loss = 0.20 from its lossless valueη lossless = 0.13, again consistent with the characterized intrinsic quantum inefficiency ofη loss = 0.25 in Ref. [19]. It is worth noting that the presented quantum inefficiency valuesη of the Floquet TWPA design are on the same order but not optimal at each loss tangent: for instance, one can leverage the amount of coherent (sideband) and incoherent (dielectric) loss and optimize the net quantum efficiency accordingly with a carefully chosen device length and nonlinearity profile. Theη of the Floquet scheme rapidly | 3,069,240 | 248060231 | 0 | 16 |
diminishes with a smaller tan δ and eventually approaches approximately 10 −4 , which is limited by the small impedance mismatch and the finite ramp rate of I pn (x) in Fig. 4(f). Floquet mode JTWPAs fabricated with a typical qubit fabrication process are predicted to have a quantum efficiency on the level of η/η ideal > 99.9% (η < 10 −3 ), demonstrating the practicality of our proposed Floquet scheme. Finally, we discuss the directionality and the prospect of directly integrating a Floquet mode TWPA on chip. In a typical superconducting quantum experiment setup, the preamplifier (JPA or JTWPA) in the measurement chain is only indirectly connected to the device under test via a commercial isolator or circulator to prevent re- flections from dephasing the qubits or causing parametric oscillations in the amplifier. Such insertion loss occurring before the preamplifier will degrade the measurement efficiency appreciably and a directional integrated quantumlimited preamplifier is therefore essential for approaching near-perfect full-chain measurement efficiency. While TWPAs are in principle directional, existing TWPAs cannot fulfill this promise due to their non-negligible reflections, as also evidenced in Fig. 4(g). It is worth noting that for well-impedance-matched amplifiers, the major obstacle is in fact the nonlinear forward-backward mode coupling, which is properly captured by the off-diagonal block matrices K 12 (x) and K 21 (x) constituting K(x) in Eq. (2). In Figs. 5(b) and 5(c), we compare the nonlinear impedance and the signal-reflection spectrum of both the conventional homogeneous scheme and the Floquet mode scheme at tan δ = 0. At | 3,069,241 | 248060231 | 0 | 16 |
nonzero loss tangent, backward propagation gets further attenuated and the signal reflection decreases correspondingly. We observe that the signal reflection in the conventional homogeneous scheme is significantly worse than the Floquet scheme even at near-identical and near-ideal impedance matching conditions. In contrast, the Floquet mode TWPA minimizes the nonlinear coupling contribution due to the adiabatic Floquet mode transformation and achieves < −25 dB reflection over the entire amplifying bandwidth. To support the claim that the signal reflection of a Floquet mode TWPA is near ideal and limited by impedance mismatch at the boundaries, we simulate the Floquet scheme again using the exact same configurations but manually disabling the nonlinear forward-backward couplings by setting K 12 (x) = K 21 (x) = 0 m , which is plotted in gray in Fig. 5(c). Indeed, the signal reflection of this "nonlinearly forward-backward decoupled" hypothetical device is almost identical to the actual Floquet TWPA (red) as expected. To operate TWPAs as true directional amplifiers and minimize the backaction on devices under test, an additional hurdle to overcome is to minimize the pump reflection and signal reflection at the same time. The pump reflection is largely affected by the mismatch between the port impedance and the pump nonlinear impedance of the TWPA at the boundaries. Floquet mode TW-PAs are advantageous in achieving this goal: whereas the strong pump tone sees a different dispersion and nonlinear impedance than the weak signal and idler tones due to self-phase rather than cross-phase modulations in both amplifier designs, the Floquet TWPAs further minimize this | 3,069,242 | 248060231 | 0 | 16 |
discrepancy at the boundaries due to the significantly reduced nonlinearity there. The pump reflection is evaluated to be S 11,pump ≈ −48.4 dB, using the same parameters in calculating Figs. 4 and 5 (for details, see Appendix E 4). The improved directionality of Floquet-mode TWPAs, as well as the insensitivity to the out-of-band impedance environment described in Section V, suggests that Floquet mode TWPAs are still favorable to conventional TWPA designs even when existing fabrication processes (tan δ ≈ 10 −3 ) are used and the quantum efficiencies of these amplifier designs are similar. V. INSENSITIVITY TO OUT-OF-BAND IMPEDANCE ENVIRONMENT In this section, we discuss the impact of a nonideal impedance environment on the performance of the proposed Floquet mode TWPAs and conventional TWPAs. In the calculations above, the port impedance is assumed to be Z 0 = 50 Ω over the entire frequency range. In practice however, qubits and TWPAs often see a quite different impedance environment at frequencies higher than 16 GHz due to wirebonds, attenuators, connectors, circulators, and other microwave components that are not optimized at those out-of-band frequencies. Alternatively, one might be tempted to intentionally engineer the impedance environment of a conventional TWPA to filter out the higher-frequency sidebands. In general, the multimode, quantum input-output theory framework presented in this work can model an arbitrary nonideal impedance environment and its effects on TWPA performance using a frequency-dependent port impedance Z 0 (ω). For both of the specific scenarios discussed above, we can use a simple stepwise impedance model of Z 0 (ω) | 3,069,243 | 248060231 | 0 | 16 |
= 50 Ω for 0 ≤ |ω/(2π)| ≤ 16 GHz and Z 0 (ω) = Z ob otherwise to emulate the behavior of large impedance mismatch outside the target frequency range. Here, we use Z ob to denote the out-of-band impedance. Figure 6 compares the numerically simulated performance of a conventional TWPA and our proposed Floquet TWPA at several out-of band impedance values Z ob = 50 Ω, 100 Ω, and 1000 Ω, respectively, with all other settings kept equal as described in Appendix A and Section III. Z ob = 100 Ω corresponds to an outof-band linear reflection of approximately −10 dB due to, for example, the high-frequency behavior of wirebonds, whereas Z ob = 1000 Ω emulates both the typical out-of-band response of commercial isolators and circulators and the intentional filtering of sideband frequencies > 16 GHz. As the out-of-band mismatch increases, we see that the gain ripples of the conventional TWPA increase drastically. In the case of Z ob = 1000 Ω, filtering out higher-frequency sidebands does help decrease the quantum inefficiencyη down to approximately 1% at gain-ripple peaks, but at the same time increasesη to as large as approximately 60% at gain-ripple troughs. The large variation in both gain values and quantum efficiency with respect to frequency thus makes this scheme unattractive for applications requiring broad and uniform gain and quantum efficiency. In stark contrast, the proposed Floquet TWPA is significantly less susceptible to changes in out-of-band impedance environments as evidenced by the minimal changes in its gain profiles. We see that | 3,069,244 | 248060231 | 0 | 16 |
the quantum inefficiencyη remains below 1% at all frequencies and is still superior overall. The drastic difference between the responses of a conventional TWPA and a Floquet TWPA here can be explained by the efficient sideband suppression of Floquet TWPAs. While the round-trip loss of sidebands decreases significantly under poorly controlled or intentionally engineered low-pass out-of-band impedance environments, the round-trip gain of a Floquet TWPA remains minimal and significantly smaller than that in a conventional TWPA, making the undesir-able parametric oscillations of sidebands much less likely in Floquet TWPAs. This suggests that Floquet TWPAs have the additional practical advantage of being insensitive to the out-of-band impedance environment, which could drastically reduce the design complexity and control requirements of the experimental setup. In addition, it also implies that low-loss Floquet TWPAs can be realistically implemented with distributed capacitors and resonators in a high-quality qubit process with minimal sacrifice in performance (for details, see Appendix E 5). This is because aside from parasitics, such distributed capacitive elements differ slightly from their ideal lumpedelement counterparts on dispersion and impedance at high frequencies (see Appendix B 2), to which the Floquet TWPAs are shown to be much less insensitive. VI. CONCLUSION In conclusion, we propose an adiabatic Floquet mode scheme that allows for both high gain and near-ideal quantum efficiency over a large instantaneous bandwidth. In the cQED platform, we show in calculations that a Floquet mode JTWPA can achieve > 20 dB gain, 1 − η/η ideal < 10 −3 , and < −20 dB reflection over 6.5 GHz | 3,069,245 | 248060231 | 0 | 16 |
of instantaneous bandwidth, using a fabrication process with tan δ ∼ 10 −6 , typical of qubit fabrication. Crucially, the proposed Floquet mode TWPAs are directional and can thus be directly integrated on chip, potentially leading to near-perfect full-chain measurement efficiency. In addition, their insensitivity to the out-of-band impedance environment, due to sideband suppression, significantly mitigates gain ripples, thus reducing parametric oscillations and instability. We expect this general Floquet mode amplifier paradigm to have far-reaching applications on amplifiers in various platforms and pave the way for scalable fault-tolerant quantum computing. Table I lists the circuit parameters used to calculate Fig. 3 and the traces corresponding to the conventional TWPA design with f c = 73 GHz in Figs. 3 to 5. They are similar to those in Ref. [19], except here we assume zero loss or tan δ = 0. Each unit cell has only one junction. Table II lists the circuit parameters used to calculate Figs. 4 to 6. Each unit cell has two identical junctions in series. In both tables the length of a single unit cell is denoted by a. Appendix B: MultiMode Quantum Input-Output Theory Lagrangian and Hamiltonian We consider the unit cell design of a generic resonantly phase-matched Josephson traveling-wave parametric amplifiers (JTWPAs) shown in Fig. 7. It is similar to but more general than those in [18,19], as we allow the circuit parameters to have an arbitrary spatial dependence denoted with the subscript n to model Floquet mode TWPAs. The circuit Lagrangian can be expressed as where in the last | 3,069,246 | 248060231 | 0 | 16 |
step we take the continuum approximation Φ n+1 − Φ n ≈ aΦ x (x)| x=na and n a ≈ dx/a, assuming that the unit cell length a is much smaller than the characteristic wavelength of the system. Here, we also use the subscript notation Φ x(t) ≡ ∂Φ/∂x(t) to denote partial derivatives with respect to x(t). To simplify notations, we introduce the normalized units and dimensionless variables is the reduced flux quantum and L J0 = φ 2 0 /E J0 is the reference junction inductance of choice (For Floquet mode TWPAs, we choose the reference to be at the center where the effective drive amplitude is maximum). The Lagrangian in Eq. (B1) can now be equivalently expressed with the dimensionless variablest,x,φ(x,t),ψ(x,t) as in which the dependence ofφ,ψ onx,t is implicitly assumed, and are the dimensionless parameters describing the spatial profile of the circuit elements. We note that implicit in Eq. (B3) and Appendix B 1 is the assumption that the plasma frequency ω J (x) = 1/ L J (x)C J (x) = 1/ √ L J0 C J0 of all the junctions remains constant, as was also assumed in the main text. C J (x) thus scales with junction critical current I 0 (x) and is accounted for by the prefactor µ(x) of the junction capacitance termφ 2 xt in Eq. (B3). This can be conveniently implemented to control the junction properties in experiment by changing the junction area. Unless otherwise noted, we will work entirely in the normalized unit from now on | 3,069,247 | 248060231 | 0 | 16 |
and omit all tildes for brevity. Following similar procedures in [59], we identify the dimensionless node fluxes φ(x, t) and ψ(x, t) as canonical coordinates, and the corresponding canonical momenta π φ (x, t) and π ψ (x, t) are therefore Applying the Legendre transform, we arrive at the Hamiltonian where in the last step we perform integration by parts on the term −φ t d(βµ(x)φ xt )/dx and produce the additional constant boundary terms, which will be dropped from now on for analyzing dynamics inside the TWPA. We quantize the system by promoting the variables to operators such that they obey the commutation relations (B9) Quantum Spatial Equation of Motion The quantum spatial equations of motion can thus be derived from the Heisenberg equations of motion: To make further progress, we make the stiff-pump approximationφ(x) → φ p (x) +φ(x), in which φ p (x) is a classical number that is solved independently from the dynamics of signals and sidebands. Moreover, we neglect the generation of the pump higher harmonics 3ω p , 5ω p , . . . and solve for the fundamental frequency pump consistently in the form of dφ p /dx = A px0 (x) sin ω p t − x 0 dx k p (x ) . It should be pointed out that although here we neglect the higher harmonics of the pump, higher order nonlinear processes 4WM, 6WM, . . . from the higher order junction nonlinearities mediated by the fundamental frequency pump are all accounted for and treated appropriately. | 3,069,248 | 248060231 | 0 | 16 |
After performing Fourier transform and cross-eliminatingψ(x, ω), we finally arrive at the single-variable equation of motion in the flux basis where J 2n (z) is the Bessel function of the first kind of order 2n, and accounts for the effect of the phase matching resonators (PMRs) and acts as an effective frequency-dependent capacitor. Notice that the cross-elimination is only valid when the frequency ω is away from the resonance ω rt [59]. From the left-hand side of Eq. (B12), we also see that the coupling strength of PMRs is described by γ(x)/ν(x) = C c (x)/C g (x), which can be made constant to maintain a similar phase matching condition across the device. For an injected signal at frequency ω s = ω 0 , the only frequency components it can couple to are ω n = ω s + 2nω p , where n is any integer. In practice however, n cannot be an arbitrarily small(negative) or large(positive) due to the restrictions of the junction plasma frequency and the transmission line cutoff frequency (from the discreteness of lumped-element transmission lines). We can therefore truncate the number of frequency components coupled to the signal to a finite number m = n max − n min + 1 and define a flux operator vector as We can now rewrite Eq. (B12) as a matrix equation in block matrix format: in which the normalized m × m frequency, inductance, and capacitance block matrices are defined as W = diag(· · · , ω n , · · · | 3,069,249 | 248060231 | 0 | 16 |
), (B16) C(x) = diag(· · · , ν(x) + γ(x)α r (ω n ), · · · ), and (B17) The last term in L −1 (x) is a Toeplitz matrix, and we use the notations J 2n ≡ J 2n (A px0 (x)) and θ 2n = θ 2n (x) ≡ exp −i2n x 0 dx k p (x ) for readability purposes. Note that as long as no ω n is outside of cutoff frequency or fall in between the resonant bandgap [ω r , ω rt ] and the pump current is below the junction critical current, W, C(x), and L −1 (x) are all positivedefinite matrices, and therefore the inverse of L −1 (x) or −1 exists and is well-defined. Although here we only consider the case of linear capacitors and PMRs connecting nodes to ground as described by a diagonal C(x), it is worth noting that our presented input-output quantum framework is general and capable of modeling lossless nonlinear capacitors or any blackbox design described by a diagonal admittance matrix Y(x), with Y nn (x) = Y (ω n ) being the admittance at frequency ω n . In this case, the corresponding block capacitance matrix becomes C(x) = −iW −1 Y(x)/C g0 . As an example, the diagonal capacitance matrices for a distributed coplanar stub capacitor and a λ/4 transmission line resonator are C nn (x) = tan(k tlr l tlr )/(ω n C g0 Z tlr ) and C nn (x) = − cot(k tlr l tlr )/(ω | 3,069,250 | 248060231 | 0 | 16 |
n C g0 Z tlr ), respectively. Here, k tlr , Z tlr , and l tlr are the wavevector, characteristic impedance, and physical length of the transmission line resonators, respectively, and here W and ω n are in the unnormalized frequency unit (rads). We now define the diagonal nonlinear impedance matrix of the TWPA (B19) where L nn and C nn denote the n-th diagonal elements of the two matrices. As can be observed later in the boundary condition calculations, the diagonal element Z nn (x) indeed represents the effective nonlinear impedance of mode n. Finally, applying the transformation (B20) we arrive at the field ladder operator basis equation of motion d dx in which are the m×m multimode coupling matrices that describe the forward-forward, backward-backward, forwardbackward, and backward-forward interactions. Notice that here we did not apply the usual slowlyvarying envelope approximation (SVEA) to reduce the equation of motion in the flux basis to the first order. Going beyond the SVEA allows us to capture the interactions between the forward and backward modes, model the reflection due to impedance mismatch at the boundaries, and crucially to conserve the bosonic commutation relations without making additional ad hoc approximations, such as in Ref. [60]. Appendix C: Boundary Conditions and Input-Output Theory Assuming the linear transmission lines ports at x = 0 and x = L to be semi-infinite and have inductance and capacitance per unit length of l l and c l , we can write the Lagrangian of the extended system as in whichl l = | 3,069,251 | 248060231 | 0 | 16 |
(l l ·a)/(φ 2 0 /E J0 ) andc l = (c l ·a)/C g0 are the dimensionless inductance and capacitance parameters of the transmission line ports, and the extended Lagrangian L full is piece-wise smooth. The continuity of flux φ(x) and the Lagrange's equations at the boundaries x = 0 and x = L yield the boundary conditions which can be interpreted as the flux (voltage) and current continuity conditions at the boundaries. Performing the stiff-pump-approximation, going into the frequency basis and applying again the transformations in Eq. (B20), we obtain the linearized ladder operator boundary conditions where the diagonal and off-diagonal matrices are in which Z 0 = l l /c l is the characteristic impedance of the input/output transmission line, and we use the notation Z(x) = Z 1/2 (x) and 1/ Z(x) = Z −1/2 (x). To formulate the input-output theory, we denote the input and output operator vectors as Equations (2) and (C3) constitute a two-point boundary value problem and can therefore be numerically solved to obtain the input-output relation in Eq. (4), with S 0 being the solution to the sourceless system(i.e., Appendix B 2 without the last term F ± (x)) and S n (x) being the Green's matrix solution to the system driven by a single point source F ± (x) and satisfying the boundary conditions Eq. (C3). In the lossless model where S n (x) = O 2m is zero, the numerically solved S 0 preserves the bosonic commutation relations as expected. One can also check that | 3,069,252 | 248060231 | 0 | 16 |
in the full loss model, the numerically solved scattering matrices S 0 and S n (x) together preserve the bosonic commutation relations at the output. Appendix D: Quantum Loss Model As illustrated in Fig. 2(a), dielectric losses can be modeled quantum-mechanically using a series of lossless transmission line ports, whose frequency-dependent scattering parameters are determined by the loss rate Γ(x). Similar to the time-domain Langevin equations, the effect of dissipation and its associated fluctuation on both the forward and backward waves can thus be incorporated into the lossless spatial equation of motion Appendix B 2 to get Eq. (2) in the main text. The phase factors in front ofF ± (x) are arbitrary and are chosen to be 1 for convenience [51], as they do not affect the quantum statistics of the outputs. In the case of a homogeneous TWPA driven with a constant pump, A px0 (x) = A px0 and θ 2n (x) = θ 2n = exp(−i2nk p x). Therefore, the multimode coupling matrix is periodic and has a period of x T = π/k p . We can therefore apply the Floquet theory to analyze the system. Denote the unique frequency-basis transfer matrix solution of Appendix B 2 to be Π(x), such that which is an initial value problem and can be solved numerically. The Floquet theorem states that the Π(x) can be written in the form of [53] Π where the 2m × 2m matrix P(x) has the same period x T as K(x), P(0) = I is the identity matrix, | 3,069,253 | 248060231 | 0 | 16 |
and Q is a constant 2m × 2m matrix that can be obtained from the monodromy matrix Applying the transformation B(x) = P −1 (x) A(x), Appendix B 2 can be now rewritten in the form of a constant dynamic matrix With the eigendecomposition of Q to be Q = VΛV −1 , where Λ = diag(· · · , r α , · · · ), and the columns of V are the corresponding set of normalized eigenvectors, we can therefore transform from the frequency basis A(x) into the Floquet basis Q(x) using We can gain insights of Eq. (E5) by applying Eq. (E1) to it: which shows that Floquet modes Q(x) are decoupled from each other and each propagates with a distinct dynamic factor r α , which are also often referred to as the Floquet characteristic exponents. Figure 8 shows the spatial dynamics of the system in the Floquet basis, with each curve representing the case when a specific Floquet modes is injected at x = 0. As expected, when only a single Floquet mode is injected, it does not generate or couple to other Floquet modes. FIG. 8. Spatial dynamics of Floquet modes in conventional TWPAs. Each curve describes a individual system response in the Floquet basis when a single amplifying (Qa, blue), deamplifying (Qa, red), or any of the stable (gray) Floquet mode is injected at x = 0 respectively. We can now analyze the Floquet modes using Eq. (E5). In Fig. 9 we plot the frequency mode decomposition of three | 3,069,254 | 248060231 | 0 | 16 |
Floquet modes as a function of the dimensionless drive amplitude I pn = I p /I 0 . Figure 9(a) shows the decomposition of the amplifying Floquet modeQ a , which is the same as Fig. 3d in the main text. From the decomposition of the deamplifying Floquet modeQ d in Fig. 9(b), we see that passing the bifurcation point I pn ≈ 0.02 the magnitude of the frequency mode decomposition forQ a andQ d are exactly the same and only differ in the relative phase between the frequency components. Therefore,Q a andQ d can be understood as the squeezing and antisqueezing quadratures mostly composed of the signal and idlers. Figure 9(c) corresponds to the frequency mode decomposition of a stable Floquet mode which iŝ a + 1 -like. At high I pn , more signal and idler components are mixed in as expected. Gain Scaling In this section, we provide further details about gain scaling on Floquet mode TWPAs. The broadband signal gain of Floquet mode TWPAs can be similarly extended either by increasing the effective pump strength I pn = I p /I 0,min or with a longer device length. In Fig. 10 we plot the performance scaling of a Floquet TWPA at 6 GHz with respect to driving pump strength and device length, respectively. We note that in both scenarios the signal reflection scales with signal gain due to finite reflection at boundaries. Similar to what was described in Section II on conventional TWPAs, adjusting the effective pump strength allows the signal gain | 3,069,255 | 248060231 | 0 | 16 |
of Floquet mode TWPAs to be increased in situ at the cost of larger quantum inefficiencyη, except such increase is much less pronounced for Floquet TWPAs. This is not surprising, as this degradation in quantum efficiency results from an increase in effective pump strength and therefore sideband coupling, which Floquet TWPAs efficiently suppress. Increasing signal gain with a slightly longer device length, on the other hand, provides the advantage that the quantum efficiency remains similar at higher gain. Furthermore, the increase in device length to reach a higher gain is modest: this is achieved by modifying the nonlinearity profile and slightly extending the center region near which the gain coefficient is largest (e.g. see Fig. 3(e)). For instance, using the same parameters in the main text, ≥ 30 dB gain can be achieved either by driving the same Floquet TWPA design slightly harder at I pn ∼ 0.635 (6% increase), or by a longer device with 2, 100 cells (5% increase). Pump Reflection Here, we provide further details on the calculation of pump reflection discussed in Section IV. In line with the stated assumptions in Appendix B 2, we express the first order derivative of pump flux in the sinusoidal form of φ p,x (x) = A px0 (x) sin ω p t − x 0 dx k p (x ) . In Floquet mode TWPAs, the higher pump harmonics 3ω p , 5ω p , . . . can be neglected to a very good approximation, because pump harmonic generation is minimal at the start | 3,069,256 | 248060231 | 0 | 16 |
of Floquet mode TWPAs where nonlinearity is minimal, and the pump is adiabatically transformed through the center region where the nonlinearity is large. Considering only the fundamental pump frequency ω p , substituting in the expression of φ p,x (x), performing Fourier transform on the equations of motion Eqs. (B10) and (B11), and taking derivative with respect to x on both sides, we arrive at (E7) in which we define the effective pump capacitance as c p (x) ≡ c(ω p ) = ν(x) + γα r (ω p ). Because the spatial variation of nonlinearity and dispersion are slow and adiabatic in Floquet TWPAs, we make the approximations µ x (x)/k p (x), µ xx (x)/k p (x) 2 µ(x), and A px0,x (x)/k p (x), A px0,xx (x)/k p (x) 2 A px0 (x) analogous to [18]. Expanding out the derivatives in Eq. (E7) and applying the above approximations, we arrive at From Eqs. (E7) and (E8), we see that the effective inductance and capacitance (normalized by L J and C g0 respectively) the pump tone sees are 1/ µ(x)(2J 1 (A px0 (x))/A px0 (x) − βω 2 p ) and c p (x) respectively. We also note that with Bessel function expanded to third order, 2J 1 (A px0 (x))/A px0 (x) ≈ 1 − A px0 (x) 2 /8 and Eq. (E8) recovers the usual self-phasemodulation expression of [18] in which higher than fourth-order junction potential terms are ignored. The nonlinear impedance the pump tone experiences is thus Furthermore, within the | 3,069,257 | 248060231 | 0 | 16 |
adiabatic approximation, φ p,x can be related to the effective pump strength I pn (x) as where I 0 (x) is the junction critical current at x, and I p,in is the physical input pump current (again neglecting coupling to the higher harmonics of pump). The dimensionless pump flux amplitude A px0 (x) can thus be numerically solved from the effective pump strength I pn (x) using Eq. (E10). Finally, defining the interface reflection coefficients r 01,p = (Z p,nl (0) − Z 0 )/(Z p,nl (0) + Z 0 ) and r 12,p = (Z 0 − Z p,nl (L))/(Z 0 + Z p,nl (L)), we can thus estimate pump reflection in the Floquet TWPA by Using the same parameters as in Figs. 4 and 5 and Appendix A, the pump reflection at 7.875 GHz is evaluated to be ∼ −48.4 dB. Performance with Distributed Phase Matching Resonators Heretofore we have assumed ideal lumped LC phase matching resonators in analysis. We now discuss the alternative of employing distributed transmission line resonators (TLRs) [20] and its effect on the performance of Floquet mode TWPAs. Here we choose to implement the distributed phase matching resonators using a coupling capacitor C c (x) and λ/4 TLRs shorted to ground on the other end, because at low frequencies they can be approximated as parallel LC resonators. Following the procedure presented in Appendix B 2, we can write the diagonal elements of the resulting capacitance matrix (including also the parallel ground capacitance C g (x)) as C nn (x) = | 3,069,258 | 248060231 | 0 | 16 |
ν(x) + 1 iω n C g0 1 iωnCc(x) + iZ tlr tan(k tlr l tlr ) = ν(x) + γ(x) 1 − ω n C c (x)Z tlr (x) tan(k tlr l tlr ) , in which the definitions follow those in Appendices B 1 and B 2, except here ω n is in the unnormalized frequency unit (rads). In Fig. 11 we compare the numerically simulated performance of Floquet TWPAs implemented with lumped element LC resonators and with distributed λ/4 transmission line resonators (TLRs), respectively. The parameters of the Floquet TWPA with LC resonators are the same as those used in Figs. 4 and 5, and the parameters of the Floquet TWPA with TLRs are almost identical except with reduced spacing=1 and max[C c (x)] = C c0 = 5 fF to match the coupling strength of both resonators. Moreover, we assumed a typical phase velocity v ph,tlr = ω/k tlr (ω) = 1.3 × 10 8 m/s for TLRs, and for numerical evaluation convenience we vary the characteristic impedance Z tlr (x) between 50 Ω and ∼ 300 Ω such that Cc(x)Z tlr (x) remains constant. As expected, the distributed nature of TLRs results in different dispersion and impedance of sidebands at higher frequencies. We observe that the gain and quantum efficiency are similar over most of the band. The additional features near ω r for the Floquet TWPA with TLRs are due to the modified dispersion of the sideband near 3ω r phase matching the corresponding sideband coupling process. The weak dependence | 3,069,259 | 248060231 | 0 | 16 |
of performance on resonator implementation details and higher frequency dispersion showcases the robustness and practicality of our proposed Floquet mode TWPA design and makes experimental realization using a low-loss qubit fabrication process feasible. Parameter Variation We now discuss the effects of non-ideal parameter variations to the Floquet TWPA performance. Specif- ically, we consider the case of spatial junction critical current variation that could result from fabrication nonuniformity. We model junction variation with an effective critical current profileĨ 0 (x) ≡ I 0 (x)(1 + σ Ic˜ (x)), where I 0 (x) is the ideal critical current defined in Section III, σ Ic is the standard deviation, and˜ (x) is a continuous Gaussian normal random variable with mean ˜ (x) = 0 and variance ∆˜ (x) 2 = 1. Figure 12 shows the typical performance of the Floquet TWPA design in the main text under junction variation σ Ic = 0%, 2%, 5%, and 10% respectively. The junction critical current variation primarily impacts the reflection, S 11 , and the drastic increase is caused by the random mismatch and linear reflections between adjacent unit cells. Whereas the directionality objective puts stringent requirements of sub-percent variation on junction uniformity, we observe that both the gain profile and quantum efficiency of Floquet TWPA are robust against up to few percent variations and only start to deteriorate rapidly after σ Ic > 5%. ≤ 5% junction variation has been readily achieved and reported [61,62]. | 3,069,260 | 248060231 | 0 | 16 |
Molecular dynamics simulation in virus research Virus replication in the host proceeds by chains of interactions between viral and host proteins. The interactions are deeply influenced by host immune molecules and anti-viral compounds, as well as by mutations in viral proteins. To understand how these interactions proceed mechanically and how they are influenced by mutations, one needs to know the structures and dynamics of the proteins. Molecular dynamics (MD) simulation is a powerful computational method for delineating motions of proteins at an atomic-scale via theoretical and empirical principles in physical chemistry. Recent advances in the hardware and software for biomolecular simulation have rapidly improved the precision and performance of this technique. Consequently, MD simulation is quickly extending the range of applications in biology, helping to reveal unique features of protein structures that would be hard to obtain by experimental methods alone. In this review, we summarize the recent advances in MD simulations in the study of virus–host interactions and evolution, and present future perspectives on this technique. INTRODUCTION Proteins fluctuate spontaneously in solution (Ishima and Torchia, 2000). Accumulating evidence indicates that such fluctuations play key roles in the specific functions of proteins, such as catalytic reactions of enzymes (Nicholson et al., 1995;Lu et al., 1998;Eisenmesser et al., 2005;Abbondanzieri et al., 2008), interactions with other biomolecules (Thorpe and Brooks, 2007), and biomolecular motors and pumps (Astumian, 1997). Multiple experimental methods are available to characterize the protein dynamics (Figure 1). However, it is usually difficult to delineate motions of proteins at an atomic scale. MD SIMULATION IN BIOLOGY | 3,069,261 | 18940695 | 0 | 16 |
OUTLINE Molecular dynamics (MD) simulation is a computational method to address the above issue (Figure 1) Dror et al., 2010). This technique enables us to calculate movements of atoms in a molecular system, such as proteins in water, by numerically solving Newton's equations of motions (Karplus and Petsko, 1990;Adcock and McCammon, 2006). In a simple molecular system, all atoms and covalent bonds connecting the atoms are assumed to be the charged spheres and springs, respectively. Parameters of mathematical functions describing the potential energy of a system, termed the "force field," are set to simulate the movements of atoms and molecules. Frequently used force fields for proteins, such as the "AMBER" (Pearlman et al., 1995;Case et al., 2005) and "CHARMM" (Brooks et al., 2009) force fields, have the formulae of covalent bonds, angles, dihedrals, van der Waals, and electrostatic potentials. PERFORMANCE AND CONSISTENCY WITH EXPERIMENTAL DATA Application of MD simulation in the field of protein chemistry was first reported in 1977 (McCammon et al., 1977). Since then, the performance of this technique have been quickly improved quantitatively and qualitatively along with the rapid advances in hardware and software on biomolecular simulation (Lindorff-Larsen et al., 2012). The results of MD simulation are critically influenced by the force fields (Lindorff-Larsen et al., 2012). The qualities of parameters in the force fields, especially for dihedrals and electrostatic potentials, have been improved quantitatively and qualitatively over time by introducing improved approximation to the quantum ground-state potential energy surface. Recently, eight different protein force fields were evaluated on the basis of the | 3,069,262 | 18940695 | 0 | 16 |
consistency of simulations with the NMR data (Lindorff-Larsen et al., 2012). The study demonstrates that the most recent versions, while not perfect, provide results that are highly consistent with the experimental data (Lindorff-Larsen et al., 2012). In addition, explicit introduction of effects of the solvation has contributed to the qualitative improvement for the precision and performance of MD simulations (Adcock and McCammon, 2006). MD IN STRUCTURAL BIOLOGY MD simulation currently allows us to investigate the structural dynamics of proteins on timescales of nanoseconds to microseconds, and will probably allow investigation to milliseconds in the future (Figure 1) Dror et al., 2010). This technique is widely used in the field of structural biology (Karplus and McCammon, 2002;Karplus and Kuriyan, 2005;Dodson et al., 2008). First, MD simulation is useful for refining the experimentally determined three-dimensional (3-D) structures of proteins (Autore et al., 2010;Ozen et al., 2011). Second, MD simulation is beneficial for constructing previously undescribed 3-D structures of proteins in combination with homology modeling techniques Sanchez et al., 2000;Baker and Sali, 2001), when a reported structure of a homolog is available. Third and most importantly, MD simulation provides a unique tool to address the structural dynamics of proteins, i.e., the time evolution of conformations in solution, at timescales of nanoseconds to microseconds Dror et al., 2010). The structural snapshots obtained during MD simulation are helpful for depicting the unique structural features of proteins (Karplus and McCammon, 2002;Karplus and Kuriyan, 2005;Dodson et al., 2008). MD SIMULATION IN VIROLOGY To date, MD simulations have been applied in a range of | 3,069,263 | 18940695 | 0 | 16 |
virus researches, as shown in the following sections. NEUTRALIZATION ESCAPE AND CELL TROPISM SWITCHING OF HIV-1 MEDIATED BY AN ELECTROSTATIC MECHANISM It is very important to clarify how viruses evade neutralization antibodies in order to understand the viral life cycle and evolution, and to develop vaccines. MD simulation is used to address this issue as it pertains to human immunodeficiency virus type 1 (HIV-1). The third variable (V3) loop of the HIV-1 envelope gp120 protein constitutes the major antibody epitopes of HIV-1 and the major determinants for the entry coreceptor use of HIV-1. By analyzing the 40,000 structural snapshots obtained from 10-30 ns of MD simulations of the identical gp120 outer domain carrying a distinct V3 loop with net charge of +3 or +7, Yokoyama and colleagues showed that the change in V3 net charge alone is sufficient to induce global changes in fluctuation and conformation of the loops involved in binding to CD4, coreceptor, and neutralizing antibodies (Naganawa et al., 2008;Yokoyama et al., 2012). Structural changes caused by a reduction in the V3 net charge via V3 mutations are tightly linked to viral CCR5 coreceptor tropism (Naganawa et al., 2008), as well as to a reduction in viral neutralization sensitivity to anti-V3 antibodies (Naganawa et al., 2008) and anti-CD4 binding site monoclonal antibodies (Yokoyama et al., 2012). These findings suggest a hitherto unrecognized mechanism, V3-mediated electrostatic modulation of the structure and dynamics of the gp120 interaction surface, for adjusting the relative replication fitness and evolution of HIV-1 (Yokoyama et al., 2012). In addition, they partly | 3,069,264 | 18940695 | 0 | 16 |
explain a virological mystery, i.e., why HIV-1 variants using CCR5, which carries a V3 loop with a lower level of positive net charge, predominantly persist before the onset of AIDS. MECHANISMS OF VIRAL ESCAPE FROM HOST DEFENSE SYSTEMS Viruses also evade host defense systems other than neutralization antibodies (Figure 2). MD simulation is used to clarify the structural basis for viral escape from host defense systems by mutations. Mutations at the 120th amino acid in the HIV-2 capsid protein play a key role in evading tripartite motif-containing protein 5α (TRIM5α), an anti-retroviral cellular protein induced by interferon, both in vivo (Onyango et al., 2010) and in vitro (Song et al., 2007). An MD simulation study has revealed that the mutations could extensively influence the conformation and fluctuation of the interaction surface of capsid proteins by altering the probability of hydrogen bond formation between helices 4 and 5 (Miyamoto et al., 2011). HIV-1 Vpu antagonizes an antiviral cellular protein termed tetherin, also known as BST-2/CD317/HM1.24, by interaction with the transmembrane (TM) domain of tetherin and subsequent degradation (Douglas et al., 2010;Kobayashi et al., 2011). An MD simulation suggests that alignment of the four amino acid residues (I34, L37, L41, and T45) on the same helical face in the human tetherin TM domain is crucial for the Vpumediated antagonism against human tetherin (Kobayashi et al., 2011). The interface structure of the tetherin TM for the antagonism was also predicted by the MD simulation of another group (Zhou et al., 2012) and experimentally confirmed by an NMR study (Skasko | 3,069,265 | 18940695 | 0 | 16 |
et al., 2012). MD simulation is also used to study the mechanisms of functional interactions between cytotoxic T lymphocyte (CTL) epitope and major histocompatibility complex (MHC) molecules (Reboul et al., 2012). An MD simulation study has revealed that a 13-mer epitope peptide from Epstein-Barr virus has the low structural flexibility in an MHC molecule that induces a CTL response but exhibits high flexibility in another MHC molecule that cannot induce a CTL response (Reboul et al., 2012). Thus, structural flexibility of CTL epitope region seems to be critical for the specific recognition by MHC molecules, and mutations that alter the flexibility may influence CTL response. There are other viral proteins and immune molecules involved in viral evasion from host defense systems (Neil and Bieniasz, 2009;Malim and Bieniasz, 2012). MD simulations should also be applicable for the studies of these molecules. STRUCTURE AND FUNCTION OF VIRAL ENZYMES Viral enzymes are essential for viral replications and thus are important targets for anti-viral drug development. MD simulations are used to study the basis of the structural dynamics that allow the viral enzyme and its drug to function properly. Viral polymerase (Pol) is essential for viral genome replication in the viral life cycle. The Pol is composed of the fingers, palm, and thumb subdomains, which form a cavity for the substrate binding, as in eukaryotic Pol (Joyce and Steitz, 1994;Lamers et al., 2006;Cameron et al., 2009). MD simulations suggest that the finger and thumb domains of HIV-1 reverse transcriptase (RT) are especially mobile among the various regions of this enzyme | 3,069,266 | 18940695 | 0 | 16 |
(Zhou et al., 2005;Kirmizialtin et al., 2012). The mobility is severely attenuated by binding of allosteric non-nucleotide RT inhibitors (NNRTIs) (Zhou et al., 2005). Interestingly, a large conformational change of RT subdomains during millisecond timescale simulations can lock the correct nucleotide at the active site but promotes release of a mismatched nucleotide (Kirmizialtin et al., 2012). Furthermore, conformational dynamics leading to opening and closing motions of the substrate binding cleft are highly conserved among four RNA Pols in the picornavirus family, despite the amino acid identity being as low as 30-74% (Moustafa et al., 2011). These findings are consistent with each other and strongly suggest that the structural dynamics of viral Pol play a key role in the polymerization. Viral protease (PR) plays a key role in viral propagation by catalyzing cleavages of viral precursor proteins (Pettit et al., 1994(Pettit et al., , 2002Steven et al., 2005). HIV-1 PR and other retroviral PRs have unique regions termed the "flaps" outside the substrate binding clefts (Dunn et al., 2002). MD simulation studies suggest that the PR flaps in HIV-1 are intrinsically mobile, undergoing conversions between the "semiopen," "open," and "closed" conformations (Hornak et al., 2006;Deng et al., 2011). This movement is severely attenuated upon placement of the substrate or PR inhibitor in the binding cleft (Karthik and Senapati, 2011), suggesting that flap movement plays a critical role in PR function. MD simulations are also used to study the structural dynamics of the substrates of viral PR. Peptides corresponding to cleavage junctions of viral precursor proteins of HIV-1 | 3,069,267 | 18940695 | 0 | 16 |
are intrinsically unstructured in aqueous solution (Datta et al., 2011;Ode et al., 2011). However, the folding preference of the junction peptides may be different among the junctions and related to the efficiency of substrate binding and cleavage reaction by PR . Furthermore, peptides at the capsid-p2 junction can adopt a helical conformation when the polarity of the environment is reduced (Datta et al., 2011). The MD simulation of PR and its substrates will help to clarify how the viral precursor is processed orderly during viral maturation. DRUG-RESISTANCE MECHANISMS Antiviral drug resistance is a major clinical problem for the treatment of virus-infected individuals (Cortez and Maldarelli, 2011;van der Vries et al., 2011). Viral resistance to antiviral drugs is primarily caused by genetic mutations that eventually lead to a reduction in the drug affinity of drug target viral proteins. MD simulations are used to examine how viral mutations cause the drug resistance at the atomic level. A reduction in the binding affinity of the PR inhibitors to HIV-1 PR can be caused by a reduction in hydrophobic interactions (Kagan et al., 2005;Wittayanarakul et al., 2005;Sadiq et al., 2007;Chen et al., 2010;Dirauf et al., 2010), reduction in electrostatic interactions (Ode et al., 2005(Ode et al., , 2006(Ode et al., , 2007aChen et al., 2010), changes in flexibility at the flap of the PR (Piana et al., 2002;Perryman et al., 2004;Chang et al., 2006;Foulkes-Murzycki et al., 2007), and changes in the shape of the inhibitor-binding pocket (Ode et al., 2005(Ode et al., , 2006(Ode et al., , 2007b. Reduction in | 3,069,268 | 18940695 | 0 | 16 |
binding affinity of the nucleotide/nucleoside RT inhibitors (NRTIs) to HIV-1 RT can be caused by a distinct conformational preference of NRTIs in the substrate/NRTI-binding site compared to normal substrates (Carvalho et al., 2006) or enhancement of ATP-mediated excision of misincorporated nucleotide analogs via increased accessibility of ATP to the terminus of extending DNA (White et al., 2004;Carvalho et al., 2007). Reduction in the binding affinity of the NNRTIs to HIV-1 RT can be attained by occlusion of the NNRTI-entry pathway (Rodriguez- Barrios and Gago, 2004;Rodriguez-Barrios et al., 2005) or restoration of the proper flexibility of the RT even with NNRTIs (Zhou et al., 2005). A change in volume of the binding site of influenza virus (IFV) M2 channel blockers has been shown to reduce the blockers' binding affinity (Gu et al., 2011;Leonov et al., 2011;Wang et al., 2011). Disruption of the proper guidance of IFV neuraminidase (NA) inhibitors into their binding pocket is proposed as a possible mechanism for the reduction in the binding affinity of the inhibitors (Le et al., 2010;Kasson, 2012). MD simulations are also used to study how the genetic differences of HIV variants around the world can influence the efficacy of antiviral inhibitors (Batista et al., 2006;Ode et al., 2007a;Matsuyama et al., 2010;Soares et al., 2010;Kar and Knecht, 2012). Thus, MD simulation will be valuable to assist in the study of drug efficacy when genetic information on the drug target proteins is available (Shenderovich et al., 2003;Stoica et al., 2008;Sadiq et al., 2010;Wright and Coveney, 2011). ANTIVIRAL DRUG DISCOVERY AND DEVELOPMENT MD | 3,069,269 | 18940695 | 0 | 16 |
simulations are used to assist in the discovery and development of antiviral drugs (Durrant and McCammon, 2011;Borhani and Shaw, 2012). MD simulations allow sampling snapshots of fluctuated protein structures, which include their short-lived conformations as well as stable conformations. This is beneficial for searching conformations of a protein on ligand-binding, since ligand-binding can stabilize conformation of a protein that is not the most stable at ligand-free state (Tobi and Bahar, 2005;Xu et al., 2008). Thus, the MD simulations are used to improve the enrichment performance of molecular docking during in silico drug screening by taking accounts of multiple docking poses (Okimoto et al., 2009). The method is also applied for identifying concealed drug-binding sites, which are apparently masked and not evident from the X-ray crystal structures, by considering the structural flexibility of proteins. For example, MD simulations have been used to find a trench adjacent to the active site of HIV-1 integrase (Schames et al., 2004). A site-directed mutagenesis study provided evidence that the trench indeed plays key roles in ligandbinding (Lee and Robinson, 2006). These findings have been used to design HIV-1 integrase inhibitors with potent antiviral effects (Durrant and McCammon, 2011). Likewise, MD simulations are used to assist in the development of antiviral drugs against IFV. Using this method, a universal cavity adjacent to the binding site of natural substrate has been reported with NA proteins of human 2009 pandemic H1N1, avian H5N1, and human H2N2 strains (Amaro et al., 2011). MD simulations were also used to construct a 3-D structure model of CCR5, | 3,069,270 | 18940695 | 0 | 16 |
a major coreceptor of HIV-1 (Maeda et al., 2008;Da and Wu, 2011). VIRION STRUCTURE It is essential to clarify the structure of virions in order to understand the mechanisms of viral infection and assembly. (Figure 3). Thus, far, this is one of the largest systems among the MD simulations reported in all biological fields. Notably, the virion with viral RNA was stable during the simulations, whereas the one without the RNA was unstable, suggesting that viral RNA plays a key role in stabilizing the STMV virion (Arkhipov et al., 2006). The study is consistent with the experimental data (Day et al., 2001) and therefore provides a set of rationale conditions for performing the MD simulation of virion. Likewise, Larsson et al. reported about 1-microsecondtimescale MD simulations of the satellite tobacco necrosis virus (STNV) (Larsson et al., 2012). Their study reproduced the biochemical phenomenon of the STNV virion in solution (Unge et al., 1986), i.e., the swelling of capsid upon Ca 2+ removal by EDTA treatment. These findings will provide a structural basis for identifying the key regulators of assembly and infections and for illustrating how they function mechanically. Although MD simulation of virions composed of very large numbers of atoms is still difficult in most cases, progress in the hardware and software for the simulation, together with the accumulation of biological and physicochemical information on virions, will help us to overcome these limitations in the MD simulation of virions. PERSPECTIVE Since the processing speed of computers is still doubling approximately every two years according to Moore's | 3,069,271 | 18940695 | 0 | 16 |
law, MD studies will be extended to simulations of larger and more complex system at longer timescales. This will then lead to a better understanding of the structures and dynamics of macromolecules involved in virus-host interactions. COARSE-GRAINED (CG) MD SIMULATIONS MD simulations of macromolecules consisting of large molecular systems, such as oligomeric proteins, macromolecular complexes, and membrane proteins in a lipid bilayer, and virions are desired to better understand viruses. However, such simulations require unrealistically long analytical times and high-performance computers at present, and thereby are still limited mostly to the small molecules Dror et al., 2010). To cope with this issue and to improve the practicability of long timescale MD simulation, a "coarse-grained (CG) MD" simulation has been developed (Merchant and Madura, 2011;Takada, 2012). The CG-MD simulation employs "pseudoatoms" that consist of several atoms in a group and calculates the movement of these "pseudo-atoms" rather than the movement of "individual atoms," thereby greatly reducing the calculation time (Merchant and Madura, 2011;Takada, 2012). CG-MD simulations have been used to study helicases of hepatitis C virus (HCV) and simian virus 40 and have successfully reproduced enzyme motions, such as "ratcheting inchworm translocation" and "spring-loaded DNA unwinding" (Flechsig and Mikhailov, 2010;Yoshimoto et al., 2010). Briefly, the ratcheting inchworm translocation is the unidirectional motion of the HCV NS3 helicase during translocation that occurs by the step size of one base per ATP hydrolysis cycle (Gu and Rice, 2010). Meanwhile, the spring-loaded DNA unwinding is the discrete steps of unwinding of DNA by the HCV NS3 helicase that occurs periodically | 3,069,272 | 18940695 | 0 | 16 |
via a burst of 3-bp unwinding during NS3 translocation consuming ATPs (Myong et al., 2007). CG-MD has also been applied to the study of the structural characteristics and stabilities of the capsid particle and virion (Figure 3). Such studies have been used to investigate small plant viruses (∼28 nanometer in diameter), such as the three satellite plant viruses STMV, STNV, and the satellite panicum mosaic virus (SPMV), as well as the brome mosaic virus (BMV) (Arkhipov et al., 2006(Arkhipov et al., , 2009, and more complex capsids such as poliovirus (Arkhipov et al., 2006(Arkhipov et al., , 2009, asymmetric, conicalshaped HIV-1 capsid particles (Krishna et al., 2010), and the immature HIV-1 virion . These studies have predicted various molecular interactions that can be tested experimentally. Thus, CG-MD may play a pivotal role in the MD study of micrometer-sized systems at millisecond timescale (Merchant and Madura, 2011;Takada, 2012) and therefore may uncover novel characteristics of the interactions in virus-host relationships. INTRINSICALLY DISORDERED PROTEINS Some eukaryotic proteins have no stable 3-D structure under physiological conditions Dyson and Wright, 2005). These proteins are referred to as intrinsically disordered, natively unfolded, or intrinsically unstructured proteins. They undergo structural transition from a disordered to an ordered state upon binding to target molecules such as proteins, DNA, and small molecules (Dunker et al., 2005;Sandhu and Dash, 2007). They are often related to the "hub proteins" that have many binding partners and control important biological processes Haynes et al., 2006;Sandhu, 2009). Interestingly, viral proteins or portions of viral proteins are often intrinsically disordered. | 3,069,273 | 18940695 | 0 | 16 |
These include genomelinked protein VPg protein of plant viruses (Grzela et al., 2008;Rantalainen et al., 2008;Jiang and Laliberte, 2011;Rantalainen et al., 2011), HIV-1 Tat (Shojania and O'Neil, 2010), and Vif proteins (Reingewertz et al., 2010), and paramyxovirus nucleoproteins and phosphoproteins (Habchi and Longhi, 2012). It has been proposed that the disordered structure is beneficial for viruses to gain multiple functions in the viral life cycle with limited genome size (Rantalainen et al., 2011;Habchi and Longhi, 2012;Xue et al., 2012). Clarifying the folding landscape of viral proteins by standard MD and CG-MD simulations may help in understanding the structural principles by which viral proteins execute multiple functions in the viral life cycle. dynamics of virus capsids described by coarse-grained modeling. | 3,069,274 | 18940695 | 0 | 16 |
Dynamical TAP equations for non-equilibrium Ising spin glasses We derive and study dynamical TAP equations for Ising spin glasses obeying both synchronous and asynchronous dynamics using a generating functional approach. The system can have an asymmetric coupling matrix, and the external fields can be time-dependent. In the synchronously updated model, the TAP equations take the form of self consistent equations for magnetizations at time $t+1$, given the magnetizations at time $t$. In the asynchronously updated model, the TAP equations determine the time derivatives of the magnetizations at each time, again via self consistent equations, given the current values of the magnetizations. Numerical simulations suggest that the TAP equations become exact for large systems. Introduction Within the mean field approximation, spin models with quenched disorder can be studied by analyzing their quenched averaged behavior or, alternatively, for a specific realization of the quenched disorder [1]. In the equilibrium case, the former type of analysis includes the replica method, while the latter one is usually formulated as naive mean field, TAP equations [2], or, more generally, a Plefka expansion [3]. These equations can be derived by calculating the free energy in a high temperature (weak coupling) expansion, with the first order calculation giving the mean field free energy, the second order the TAP free energy and so on. For non-equilibrium and kinetic spin glass models, soft spin systems were the first ones to be analyzed, using the Martin-Siggie-Rose generating functional formalism [4]. Spin glass models with hard spins were first treated in [5,6,7]. A powerful generating functional approach | 3,069,275 | 118552604 | 0 | 16 |
was then developed by Coolen and collaborators [8,9], and it forms the basis of our analysis here; see also [10,11] for reviews of the techniques used in both soft and hard spin models. However, dynamical TAP equations to describe the kinetics of order parameters for a specific realization of the disorder have been only derived for the spherical p-spin model [12] and the stationary state of the Ising spin model with asynchronous update dynamics [13]. In the same way that studying the quenched averaged kinetics of hard spin models usually involves a different approach compared to soft spin models, deriving the dynamical TAP equations for hard spin models is somewhat different from doing so for their soft spin counterparts. The aim of this paper is to develop a dynamical mean field theory that relates the dynamics of mean magnetizations, potentially time varying external fields, and the quenched couplings for two kinetic versions of the Sherrington-Kirkpatrick model, one with synchronous update, the other with asynchronous update. Using a generating functional approach, we derive the dynamical naive mean-field and TAP equations as first and second orders of a high temperature expansion, similar to the equilibrium case for these two kinetic models. In addition to the technical issues, the recent use of hard spin models with discrete states, e.g. Potts and Ising models [14,15,16,17], in reconstructing the connectivity of biological networks encourages the study of the dynamics of these models in more detail. Once the forward dynamics is described, it is possible to use the results to construct approximations | 3,069,276 | 118552604 | 0 | 16 |
at the corresponding levels for the inverse problem: finding the couplings, given the magnetizations and correlation functions. In this way, one can develop effective approximate reconstruction techniques that exploit the temporal structure of data and significantly improve the quality of network reconstruction in biological systems. In fact, the results of this paper have been recently used in two other recent papers on the inverse problem [18,19]. The paper is organized as follows. After defining the dynamical models in the following section, we derive dynamical naive mean-field equations using the generating functional for the synchronous updated model. We report the TAP equations, for which the details of the derivations are reported in the Appendices. We then numerically calculate the errors for these kinetic equations as a function of the strength of the couplings for the synchronous dynamics. Dynamical Model We consider a system of N Ising spins s i = ±1, i = 1, · · · , N and assume that its state at time t, s(t) = {s 1 (t), . . . , s N (t)}, follows one of the following dynamics: (i) Synchronous dynamics. In this case time is discretized and the probability of being in state s at time step t, p t (s), is given by This is, in other words, a Markov chain with transition probability W t . (ii) Asynchronous dynamics. In this case time is continuous and, p t (s) satisfies the following equation where the operator F i acting on s flips its ith spin. For each | 3,069,277 | 118552604 | 0 | 16 |
of these processes one can define a generating functional. For the synchronous case it takes the form of where for any quantity A defined as a function of a path (s(T ), . . . , s(0)), · · · indicates averaging over the paths taken by s(t) under the stochastic dynamics of Eqs. (1a) -(1c), i.e. and Tr ≡ The asynchronous case is similar expect that the sum over t in Eq. (3) should be replaced by an integration; see Appendix B. It is useful to rewrite the generating functional by considering θ i (t) for each spin and each time step as a free parameter, integrate over it, and make sure that the definition Eq. (1c) is satisfied by inserting an appropriate delta function. This yields where Dθ = i,t dθ i (t) and Dθθ = i,t 1 2π dθ i (t)dθ i (t). Using Eq. (4) in Eq. (6), we get where the parameter α is introduced to control the magnitude of the couplings, as will become clear later. The generating functional has the property that its derivatives with respect to ψ and h give the averages of the correlators involving the spins and auxiliary fields. In particular, defining and using Eqs. (7a) and (7b), we can define m i (t) andm i (t) as From Eq. (6), we can see that the ψ → 0 limit ofm i (t) is the expected value of the auxiliary fieldθ i (t) under the measure inside the integral of Eq. (6). It is easy to | 3,069,278 | 118552604 | 0 | 16 |
show that this average, similar to the soft spin case, is zero. The same limit for m i (t) gives us the mean magnetizations. We therefore have For a detailed discussion about these and other dynamical processes on Ising spin models see [11]. To derive the dynamical mean-field and TAP equations, one first calculates the Legendre transform of the logarithm of the generating functional of the process defined by Eqs. (1a) -(1c). In this dynamical case, the logarithm of the generating functional plays the role of the Helmholtz free energy in the equilibrium statistical mechanics while its Legendre transform corresponds to the Gibbs free energy. One then expands this dynamical Gibbs free energy around the zero couplings limit, similarly to the equilibrium case [3] and the soft spin model [12]. In the following, we do this for Ising spins up to linear order in the couplings for the synchronous update and use it to derive the dynamical mean-field equations. The details of how to proceed to the TAP for the synchronous and asynchronous dynamics are provided in the Appendices. Outline of the derivation of the dynamical equations The Legendre transform of the logarithm of the generating functional with respect to the real fields, h i , and the auxiliary fields, ψ i reads (12) where ψ and h are now treated as functions ofm and m through the following equalities Eqs. (13a) and (13b) together with the definition of Γ α in Eq. (12) imply Eqs. (9a) and (9b). Using Eq. (3) in Eq. (7b), Γ | 3,069,279 | 118552604 | 0 | 16 |
α can also be written as The idea now is that for α = 0 the generating functional and its Legendre transform can be easily calculated, as the spins will be independent of each other. For the generating functional we have and for the Legendre transform of log Z 0 we have , where h 0 and ψ 0 are the real and auxiliary fields for which Eqs. (9a) and (9b) are satisfied for given m andm at zero coupling (α = 0), i.e. This can be used to express h 0 and ψ 0 in terms of m andm as To calculate the integral on the right hand side of Eq. (14a) for α = 1, we can expand Γ α around α = 0 and eventually set α = 1. Using the fact that for the first derivative of Γ α with respect to α we have yielding The last two terms in Eq. (21) are zero because of Eqs. (9a) and (9b); hence The correlation function θ i (t)s j (t) α can also be easily calculated at α = 0 yielding where the last equality follows from Eqs. (17a)-(17b). Consequently, to first order in α, we have Using Eq. (13b) yields In the limit ψ → 0 for which Eq. (11) is satisfied, we have This is the dynamical (naive) mean-field equation for the evolution of the mean magnetization. The TAP equations can be derived in a similar way by expanding Γ α to second order in α, as shown in Appendix | 3,069,280 | 118552604 | 0 | 16 |
A. This yields the dynamical TAP equations To find the time evolving magnetizations for given external field and coupling within the TAP approximation, the above equation should be solved self consistently for m i (t + 1) at each time step. Note the form of the Onsager correction (the last term in Eq. (28)). The (1 − m 2 j ) term is evaluated at time step t, but m i is evaluated at time step t + 1. Thus (28) is a set of equations to be solved for m i (t + 1), not just a simple expression for m i (t + 1) in terms of the m j (t), as in naive mean field theory. The derivations of dynamical naive mean-field and TAP equations for the case of asynchronous dynamics defined in Eqs. (2a) and (2b) are given in Appendix B. As shown there, these equations read Numerical results To test the dynamical naive mean field (hereafter: nMF) and TAP equations (27) and (28), we ran simulations in which we simulated the process define by (1a)-(1c) for L time steps, for couplings drawn from a zero mean Gaussian distribution with variance g 2 /N (J ij is drawn independent of J ji ) and subjected to two alternative types of external field. One was a temporally constant field with a magnitude drawn independently for each spin from a zero mean, unit variance Gaussian distribution. The other was a sinusoidally varying external field. For each sample of the Js and the fields, we | 3,069,281 | 118552604 | 0 | 16 |
generated data from the system for r repeats, calculated m i (t) from these repeats, and used it in (27) and (28) to predict m i (t + 1). Finally, we calculated the mean squared errors of these predicted values The results for the two external fields used are shown below. Uniform field Fig . 1A shows the dependence of the error for predicting the magnetizations at time t + 1, given the measured magnetizations at t. Both TAP and nMF errors increase as g increases, but the error of nMF is always larger than that of TAP. Furthermore, how close to the true (r → ∞) values the measured magnetizations are systematically affects the nMF and TAP predictions: increasing r decreases the errors for all g. This can also be seen in Fig. 1B, where the errors at g = 0.3 are shown as functions of r, also for two different values of N. Sinusoidal field Figs. 1 C and D show the same thing as Fig. 1 A and B, but now the system is subjected to a sinusoidal external field with a peak amplitude of 0.1 and a period of 20 time steps. The results are qualitatively the same. For this case, we also look at the time dependence of the errors in TAP and nMF equations. Fig. 2 shows the time dependent error (i.e. the right hand side of Eq. (31) without averaging over time) versus time. For weak coupling, the error of both nMF and TAP are very small. At intermediate | 3,069,282 | 118552604 | 0 | 16 |
values of g, the error of nMF is still comparable to TAP. but fluctuating. At yet stronger couplings, the nMF prediction very rapidly becomes different from the actual measured values of the magnetizations. Discussion The TAP approach, formulated as a high temperature Taylor series expansion of the equilibrium Gibbs free energy [3], is a powerful method for studying equilibrium spin glass models. Similarly, dynamical TAP equations allow analyzing the dynamics of a single sample of a disordered system away from equilibrium. In this paper, we derived these equations for Ising spin glasses with both synchronous and asynchronous updates. The main idea behind the derivation is similar to the one used by Biroli [12] for the soft p-spin model obeying a Langevin equation, with the difference that instead of a MSR formalism, we had to use the generating functional approach of Coolen. For the pspin model the spherical condition results in the appearance of the the autocorrelation, s i (t)s i (t ′ ) , and response functions as order parameters in dynamical TAP. For the hard spin Ising model, this is not the case. The response function can, of course, be directly calculated from its definition and the TAP equations, but calculating correlations, s i (t)s j (t ′ ) , function requires a different approach. The derivation does not rely on the symmetry of the couplings and can, therefore, be applied to systems without detailed balance. For the stationary case, the TAP equations are identical to those derived for the equilibrium model with symmetric connections. | 3,069,283 | 118552604 | 0 | 16 |
This has been previously shown by Kappen and Spanjer [13] using an information geometric derivation for the stationary state of the asynchronously updated model. Numerical simulations with both a constant external field and a rapidly evolving one show that the TAP equations predict the dynamics of the individual site magnetizations very well. This may not be surprising given the fact that the model we studied here was a kinetic variant of the SK model for which the equilibrium TAP equations provide the exact picture. It is intriguing that the Onsager term in Eqs. (28) and (30) does not get the form J ij J ji (1 − m 2 j ), as would be expected from a simple reaction argument. This observation has also been made earlier by Kappen and Spanjer [13]. A naive argument showing that the true correction to the mean-field equations is of the type J 2 ij (1 − m 2 j ) is as follows. Starting from the exact equation The linear term vanishes, and using [δs j (t)] 2 = 1 − m 2 j (t) we have where in the second line we have used the mean field equation m i (t + 1) ≈ tanh(b i ). An important issue that we have left out in this paper is the expected number of solutions to the TAP equations for arbitrary couplings. It has been known for a long time that, at low temperature, the expected number of solutions of the TAP equations for the SK model with symmetric | 3,069,284 | 118552604 | 0 | 16 |
couplings is exponential in N [20]. It is also possible to calculate the number of metastable states for couplings with an antisymmetric component at zero temperature [7]. The TAP equations presented here allow extending the calculation in [20] to the type of couplings considered in [7] for non-zero temperatures. This calculation will be presented elsewhere. The equilibrium TAP equations, derived for spin glass models with symmetric couplings, can be used in deriving efficient approximations for solving the inverse problem of reconstructing a spin glass model from samples of its states [21,22]. As has been recently shown [18,19], the dynamical equations derived here can be employed for taking the reconstruction to a more powerful level, allowing for the reconstruction of systems outside equilibrium. Acknowledgment The authors thank Erik Aurell and Bert Kappen for discussions at various stages of this work. The use of computing resources at Gatsby Computational Neuroscience Unit is gratefully acknowledged. Appendix A. TAP equations for synchronous update For deriving the TAP equations, we note that The first term on the right hand side of the above equation is equal to zero. To calculate the next two terms, we use the Maxwell equations We are therefore interested in calculating Defining δs j (t) = s j (t) − m j (t) and δθ i (t) =θ i (t) −m i (t), this can be rearranged into the following form: Now it is simple to evaluate Eq. (1.4) The factors have to be paired and for the pair averages we use (1.8c) The terms containing products | 3,069,285 | 118552604 | 0 | 16 |
of two averages of the form δθδs vanish, because one pair factor has to have t ′ = t − 1 and the other has to have t = t ′ − 1, which cannot be satisfied simultaneously. This leaves Using this to calculate Γ α to the quadratic order in α, differentiating with respect tom j (t), and settingm j = 0 yields the dynamical TAP equations (28). Appendix B. Asynchronous Dynamics In the asynchronous case the generating functional takes the form and · · · now indicates averaging with respect to the distribution defined by the solution to the differential equation Eq. (2a). This solution can be written as The solution to Eq. (2.2b) can be written as The dynamical Gibbs free energy (i.e. the Legendre transform of the log generating functional) is then Appendix B.1. nMF for asynchronous update As we did for the synchronous case, we first calculate the non-interacting (α = 0) generating functional where now ψ 0 and h 0 are functions of m andm from the following equations and For nMF, we need to calculate the linear term in α. This is where the last equality follows from Consequently, up to the linear term in α, we have Using the fact that ∂Γ 0 /∂m i (t) = ih 0 i (t), we find that Together with the fact that for ψ 0 = 0, we have m i (t) = µ 0 i (t), the mean-field equation is and the non-zero contributions come from pairing the terms inside | 3,069,286 | 118552604 | 0 | 16 |
the averages. Non-zero contributions come from δθ i (t) 2 α . A correlation function of the form δθ i (t)δs j ′ (t ′ ) α is nonzero for t ′ < t but since it always appears multiplied by δs j (t)δθ i ′ (t ′ ) α , which is zero for t ′ < t, it does not contribute to the final results. We therefore have To evaluate the above expression we first note that where the last equality follows from δµ 0 j (t)/δh j (t) = 0 and The dynamical Gibbs free energy can then be written as where in the last sum [δθ i (t)] 2 0 should be considered as a function of m andm. First note that the second term on the right-hand side of Eq. (2.19) is of quadratic order in ψ in the limit ψ → 0 (from Eq. (2.18a)). Butm is linear in ψ (from Eq. (2.7b)), so this term is of second order inm and its derivative with respect tom vanishes aŝ m → 0. Thus we can discard it in finding the TAP equations. We are now interested in the following quantity where x i (t) = tanh(h 0 i (t)). Form j (t) → 0, The only term that will be nonzero on the right hand side of Eq. (2.26a) is the first, as long δx i (t ′ )/δm i (t) does not diverge as fast as or faster than 1/m asm → 0. Whether δx i (t ′ )/δm | 3,069,287 | 118552604 | 0 | 16 |
i (t) is regular in the limitm → 0 or not depends on whether the functional matrix δ(m,m)/δ(h, ψ) is regular in this limit. The latter is not singular when the generating functional is regular unless the system is at a phase transition. Assuming that this is not the case, we can ignore the last term in Eq. (2.26a). Now we can proceed the way we did in the naive mean-field case, but evaluating Γ α to second order in α. The functional derivative of Γ α with respect tom, evaluated at α = 1, gives ih: tanh h 0 i (t) can be related to µ 0 i (t) through dµ 0 and µ 0 i → m i when ψ andm → 0, yielding the TAP equations Note that these are of the same form as those for the synchronous-update model with m i (t + 1) replaced by m i + dm i /dt. | 3,069,288 | 118552604 | 0 | 16 |
Using YouTube to Improve Students’ Speaking Skill in Teaching Procedure Text Published: March 9, 2022 English has become an international language, so the mastery of English language skills is important. However, the students still face some problems that make them unmotivated and hinder speaking. Considering these situations and conditions, the teacher must address this challenge and determine the most effective strategy for involving students in the teaching and learning process, particularly in speaking. As a result, YouTube is selected as one of the options for teaching speaking in this article in order to determine how using YouTube in teaching procedure text might improve students' speaking skills. On the basis of the literature review and the application of YouTube in the classroom, it is concluded that YouTube can be beneficial to enhance students' speaking skills in teaching procedure text. INTRODUCTION English has become an international language used to develop relationships with various communities around the world and for educational purposes, including in Indonesia [1] [2]. Therefore, nowadays, the mastery of English language skills is important. The success of learning English cannot be separated from the success of learning four skills including listening, speaking, reading, and writing. However, students still face some problems that hinder them to speak. As a result, they are unmotivated to speak, as well as this certainly has an impact on their speaking achievement and performance. On the other hand, there is the advancement of technology in this era. Along with the times, especially since the COVID-19 pandemic, learning is mostly carried out by | 3,069,289 | 247371228 | 0 | 16 |
utilizing modern technology, so students can easily access technology that involves them to consistently get entertainment, personal communication, and seek information (Trinder, 2017, as cited in [3]). It means that fun learning can lead to better education. One of the best ways is to use YouTube as a platform that provides many videos that students can use to get a lot of information in a fun condition because they also watch the visualizations [3]. Considering these situations and conditions, the teacher should address this challenge and determine the best strategy for involving students in the teaching and learning process, particularly in speaking. Therefore, this paper selects YouTube as a strategy for teaching speaking. In addition, this paper focuses on material for grade IX, namely procedure text. The implementation of this strategy adapts to the three stages proposed by [4], which include the pre-task-based phase, the task cycle, and the language focus phase. According to Burke, et al. (2009) 6 and Khalid (2012) as cited in [4], YouTube is a web-based public-access platform in which videos are shared online such as through www.youtube.com, other webpages, mobile platforms, blog posts, and emails, which can be used by teachers to solve problems in teaching speaking. After watching videos from YouTube, it is expected that students should be able to generate speaking ideas, add new vocabularies, know how to pronounce them, and finally improve their speaking skills. In conclusion, the purpose of this paper is to show how using YouTube can help students enhance their speaking skills when presenting procedure | 3,069,290 | 247371228 | 0 | 16 |
text. RESEARCH AND METHOD This article is written using the literature review method involving theories related to the topic of research. Therefore, the concepts and theories used as the basis of the research are built by comparing the concepts and theories used with the existing literature. In addition, those are combined with the results of the application of the teaching method in the classroom. Finally, the result is the interpretation of the literature review and the application of the method in the classroom. RESULT AND DISCUSSION 3.1 YouTube 3.1.1 YouTube Definition Steve Chen, Chad Hurley, and Jawed Karim, three former employees of the American e-commerce corporation PayPal, founded YouTube in February 2005 [5]. It has served as an illustration of how information technology (IT) has advanced in this global age. YouTube is a video-sharing website including video and TV clips, music videos, movie trailers, and other types of content like video blogging, short original films, and instructional videos, among other things [6]. YouTube is a type of amusement in which a story is told through sound and a series of images that provide the impression of continuous movement. One of the media used to teach languages is YouTube. It is quite beneficial because it might increase student enthusiasm in studying. Students may become tired of the scenario in English class, therefore they require amusement, and YouTube is one of the most appropriate mediums in the present period. YouTube is the most popular and largest online video platform in the world, especially on the internet. Currently, YouTube | 3,069,291 | 247371228 | 0 | 16 |
users of all ages, from children to adults, can be found all over the world. Users can use the site for free to upload videos, search for videos, watch videos, debate videos, and share video clips. It is accessed by millions of individuals every day. As a result, YouTube has the possibility to be applied as a tool for learning English. Characteristics of YouTube Users can browse, upload, comment, subscribe, and join groups on YouTube, just like they can on other video-sharing sites. Viewers can see the title, content, who sent the video, the date and time of upload, and any tags that the owner has added when viewing a video. Additionally, the number of times a video has been viewed and rated by registered users is made available. A list of comparable videos has been compiled based on the video title, descriptors used, and tags. Any video can be flagged for improper content, subscribed to a content feed for a certain user or channel, or added to a user's favorites. In addition, users can offer a text or video comments on each video (Duffy, 2008, as cited in [7]). About the duration of the video, it appears that YouTube users prefer shorter films. Gill et al., 2007, as mentioned in [7], discovered that videos with long-term popularity are typically shorter, with videos of 3-5 minutes accounting for 52 percent of the All-Time Most Popular category. While for popular subjects of videos that are uploaded to YouTube, they are automatically placed into one of 15 pre-defined | 3,069,292 | 247371228 | 0 | 16 |
categories. Music, humor, entertainment, and sports routinely rank among the most viewed categories, according to research (Cheng et al., 2007;Gill et al., 2007;Madden, 2007, as cited in [7]). Madden (2007, as cited in [7]) pointed out that news appeals to people of all ages, with the exception of young adults aged 18 to 29, who choose comedy (56 percent comedy versus 43 percent news). It was also stated that educational videos are popular with people of all ages. Advantages and Disadvantages of YouTube According to [6], there are numerous benefits by using YouTube in the classroom, some of which are described here. a. YouTube is regarded as a valuable resource because it may be used both inside and outside of the class. b. It exposes students to real English. c. It encourages a more self-directed learning method and students' centered. d. It can serve as a source of incentive for students, allowing them to stay in class for extended periods of time. YouTube has the potential to be a highly effective online learning and teaching resource. g. In the classroom by using YouTube it can captures students' interest and makes language learning highly dynamic. h. Students can ask questions and provide comments about videos on YouTube. i. YouTube is not to mention as a learning tool, but it also serves to inspire students. j. Students can also use YouTube to convey what they have studied by creating a video and sharing it with the class. k. Suppose a student sees a video on YouTube that is | 3,069,293 | 247371228 | 0 | 16 |
both engaging and beneficial in English language study, he or she can quickly find out more films that are comparable to it. l. It includes authentic examples of regular English spoken by the individuals. In addition to such advantages, YouTube has some disadvantages in class, among them are as follows. a. Students may have difficulty understanding the terminology used in videos. b. For students, the linguistic level employed in the videos can be confusing. c. There may be concerns about violation of privacy. d. There could be concerns with copyright. e. Some videos may be deemed inappropriate without prior notice. f. There are no restrictions on comments. g. There is no control overusing YouTube. h. When students are looking for resources, they may come across distasteful or inappropriate content. i. YouTube can quickly divert students' attention away from the subject at hand. j. When utilizing YouTube in the classroom, there is no option to block the inappropriate content. Speaking Skill 3.2.1 Definition of Speaking Speaking is one of the four fundamental skills that students should master. Speaking considered as a productive skill. It was impossible to separate it from listening. We create the text when we talk, and it should be relevant. In the nature of communication, we may find the speaker, the listener, the message, and the feedback. Speaking is inextricably linked to pronunciation since it encourages students to learn English sounds. Some scholars have presented a number of different definitions of speaking. Speaking is a productive skill that may be directly and experimentally observed | 3,069,294 | 247371228 | 0 | 16 |
[8], but those observations are usually tinted by the precision and effectiveness of a test taker's listening skill, compromising the reliability and validity of an oral production exam. In a classroom, speaking involves interaction between teachers and students, or amongst students, depending on how the classroom is set up. Speaking has various significant traits as compared to writing and reading skills (often referred to as written language, receptive skills). Speakers rarely employ whole sentences when speaking, and their vocabulary is less specific than in written language. Making use of words in a normal voice, offering words, knowing and being able to utilize a language, expressing oneself in words, and giving a speech are all examples of speaking [9]. Speaking, according to [10], is an interactive process of meaning construction that involves both the production and reception of information. Its appearance and meaning are determined by the context in which it is used. Based on the aforementioned arguments, it can be stated that speaking is the most crucial skill to learn in order to communicate with others and explain our ideas in a way that the listener can understand. Components of Speaking a. Vocabulary Vocabulary is defined as the entire quantity of words that make up a language (together with rules for combining them), or the range of words that a person knows or uses in a trade, profession, or another setting [9]. Vocabulary means the appropriate diction which is used in communication. One cannot properly communicate or express their ideas in both oral and writing form | 3,069,295 | 247371228 | 0 | 16 |
without a suitable vocabulary. Learners who have a restricted vocabulary have another challenge to acquiring a language. As a result, language teachers should develop a thorough understanding of how to manage an engaging classroom so that students can achieve tremendous success in their vocabulary learning. Little can be communicated without grammar, and nothing can be communicated without vocabulary. b. Grammar Students must be able to create the right sentence in a conversation. Simon and Yatvin in Syahril (2011, as cited in [11]) explain grammar as follow: 1) The branch of linguistics concerned with the words' forms and structures (morphology), as well as their typical arrangement in phrases and sentences (syntax). 2) A language's system of word structures and arrangements at a specific point in time. 3) A set of guidelines for writing and speaking in a specific language. 4) A booking accommodating such instructions. 5) The way someone speaks or writes. Because of the varied transmission of speech, it should be taught through an awareness of the structural pattern. c. Pronunciation Pronunciation is highly crucial in speaking; if we do not pronounce a word correctly, it can change its meaning. According to [9], the way a language is spoken, the way a word is pronounced, the way a person speaks the words of a language is known as pronunciation. Pronunciation is concerned with the articulation of words, and it is crucial since incorrect pronunciation will cause the interlocutor difficulty in comprehending the message [9]. When students communicate, they use pronunciation to develop clearer language. The phonological | 3,069,296 | 247371228 | 0 | 16 |
process is concerned with the pieces and concepts that describe how sounds fluctuate and pattern in a language. d. Fluency In Oxford Learner's Dictionary, fluency is defined as the ability to speak a language smoothly and easily. In addition, Herlina (as cited in [9]) states that fluency refers to a person's ability to continue speaking in a natural manner. It is a term of communication. Marcel (1978, as cited in [12]) says that fluency is someone's manner of speaking by processing the suitable words and sentence patterns. Procedure Text 3.3.1 Definition of Procedure Text A text is a person's communicative act which includes sending complete written or spoken sentences separated by silence, blank space, or the sender's shift [13]. Based on his opinion, it means that the text is closely related to written language and spoken language. The researchers categorize some types of texts, and one type of text that is taught in grade IX at junior high school according to the syllabus is procedure text. Anderson (2003, as cited in [14]) explained that a procedure text is a piece of written or spoken information that instructs the reader or listener on how to perform something. Moreover, this text is aimed to provide instructions about the steps, methods, or ways of doing something [15]. Components of Procedure Text A text has three components of text, which contains social function, generic structures, and language features. It also applies to the procedure text, and its components are explained in the following paragraph. First, the social function of a | 3,069,297 | 247371228 | 0 | 16 |
procedure text is to explain how something can be done [16], as well as to give instructions and tell how to do, make, or operate something. It has three generic structures, namely aim or goal, ingredients or materials, and steps or methods. Moreover, it has the following language features: focusing on generalized human agents, using the simple present tense, and using temporal conjunctions, such as first, second, then, next, after that, finally. Several types of procedure text are usually taught to junior high school students, for example, recipes, manuals, tutorials, and tips. Teacher divides the students into several groups of four or more students. j. The Application of YouTube in the Classroom Teacher asks the students to watch the second video once more and take notes to help them adjust the content of the text so that they can start reconstructing the text in groups. k. Students are given the task of making a procedure text as the sample video they have watched. l. Teacher instructs the students to practice reconstructing the procedure text they have compiled through voice notes or video calls as their homework. That's good, but it will be better if you .... Thanks for your hard work, it's nice, but you should .... e. Teacher asks the groups to start making the video, in which they can do it at their own home or in groups (face-to-face), while still implementing the health protocols. f. Students in groups begin the process of filming the procedure text video, which has been given feedback, as | 3,069,298 | 247371228 | 0 | 16 |
creatively as possible. g. Students upload their video to one of their YouTube channels after all processes are done. Teacher rewards the best group with the best video. Sample of Material Congratulations to the best group! I have a present for the best group. Big applause for the best group! One example of students' group work is a video in the following YouTube link: https://www.youtube.com/watch?v=AV9oRyrB_3M. CONCLUSION English has become an international language, so the mastery of English language skills is important. However, the students still face some problems that make them unmotivated and hinder speaking. Considering these situations and conditions, the teacher must address this challenge and determine the most effective strategy for involving students in the teaching and learning process, particularly in speaking. Therefore, YouTube is selected as one of the options for teaching speaking in this article in order to determine how using YouTube in teaching procedure text might improve students' speaking skills. On the basis of the literature review and the application of YouTube in the classroom, it is concluded that YouTube can be beneficial to enhance students' speaking skills in teaching procedure text. | 3,069,299 | 247371228 | 0 | 16 |
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