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heavy impurity ordered phase disappears may be derived from enforcing the limit µ → ∞ in the equality λ Equation (52) yields where use has been made of the relation where F (x) is given by Eq. (37). The identity (54) allows for a convenient expression of a [and hence a * through Eq. (38)], once λ [and hence F ( a)] is known. Since the value of a * (+) has been obtained from the condition λ (0) 2 = 0 (constant temperature in the ordered phase), the expression (53) also gives the α-dependence of the shear rate in the steady USF state [18]. The quantities µ (−) th and a * (+) are indicated by arrows in Fig. 4. Consistent with the disappearance of the heavy impurity ordered phase for elastic systems, is the vanishing of a * (+) when α 22 → 1. It can also be noted that a * (+) is only a host property, and does not depend on α 12 . Finally, Fig. 5 shows that the behavior in two and three dimensions are similar. We now turn to asymmetric collisional dissipation cases, so that the light impurity ordered phase may exist even for vanishing shear rates. Such a scenario is illustrated in Fig. 6, that corroborates the analytical predictions. The boundary of the light impurity ordered phase in a shear-mass ratio phase diagram is non trivial, and indicates the existence of an interval of µ values, below µ (−) HCS , with a re-entrance feature. Indeed, starting from | 3,068,600 | 11722037 | 0 | 16 |
the ordered phase at a * = 0, and increasing a * at fixed µ, one first meets a transition from order to disorder, followed by a subsequent ordering transition. Similarly, and again for α 12 > (1 + α 2 22 )/2, the following series order → disorder → order occurs when µ is increased at fixed reduced shear rate, provided a * < a * (+) . To substantiate the phase diagram reported above, we show in Fig. 7 the shear rate dependence of the order parameter E 1 /E, for different values of the (common) coefficient of restitution and mass ratios. The light impurity ordering is seen to be enhanced by increasing the shear rate, while the reverse behavior is in general observed for heavy impurities (see the inset). The critical thresholds observed in Fig. 7 are fully compatible with those appearing in Fig. 4. Focussing next on the light impurity ordered phase, we report the order parameter variation in cases of asymmetric collisional dissipation. To this end, we return to the set α 12 = 0.9 and α 22 = 0.55 addressed in Fig. 7. For such quantities, one has µ As can be seen in Fig. 8, when µ < µ (−) HCS , the order parameter is non vanishing at small shear rates, while when µ th , ordering sets in only beyond a critical shear rate (note that µ (−) th ≃ 0.733, so that the light impurity ordered phase does exist in some shear domain, for all the | 3,068,601 | 11722037 | 0 | 16 |
mass ratios used in Fig. 8). The figure clearly shows the re-entrance of order alluded to above (see the curves for µ = 10 −2 and µ = 3 × 10 −2 , where an intermediate interval of a * values leads to a disordered phase with E 1 /E = 0. For µ = 10 −3 (dashed-dotted line), all shear rates lead to phase ordering (E 1 /E = 0), but there is a fingerprint of the reentrant behavior in the non monotonicity of the order parameter with a * . For completeness, we now show how the tracer limit is approached. For µ < µ (−) th and α 12 = α 22 , ordering occurs when the system is driven sufficiently far from equilibrium (a * > a * c ). To illustrate how the abrupt transition observed in the tracer limit is blurred by finite concentration, Fig. 9 shows E 1 /E versus the (reduced) shear rate a * for d = 3, m 1 /m 2 = 0.2, α 11 = α 22 = α 12 = 0.9 and x 1 = 10 −1 , 10 −2 and 0. It is apparent that the curves tend to collapse to the exact tracer limit result as the mole fraction x 1 vanishes. This is indicative of the consistency of the analytical results derived at x 1 = 0. Moreover, since the impurity particle is sufficiently lighter than the particles of the gas (µ = 0.2 < µ (−) th ≃ 0.414), the | 3,068,602 | 11722037 | 0 | 16 |
energy ratio E 1 /E is non vanishing in the tracer limit if a * > a * c ≃ 7.557 in the present case (see the figure). It can also be noted that at finite x 1 , the energy ratio for small shear rates is of the order of x 1 , since the temperature ratio is quite close to unity in that limit (see e.g. Fig. 2). Although we have focused our attention on the energy ratio, it is clear that similar features can be analyzed when considering other quantities. An interesting candi- date is the non-linear shear viscosity η * defined as where the shear stress P * xy = P * 1,xy + P * 2,xy . The rheological function η * characterizes the nonlinear response of the system to the action of strong shearing. In terms of p * 1 and for finite values of x 1 , the expressions of P * 1,xy and P * 2,xy are given by Eqs. (A6) and (A7), respectively. As expected, in the tracer limit (x 1 → 0) and in the disordered phase, the total shear viscosity η * of the mixture coincides with that of the solvent gas η * → η * s , where η * s is given by Eq. (C18). However, in the ordered phase, there is a finite contribution to the total shear viscosity coming from the tracer particles (see Eq. (C19)). To illustrate it, Fig. 10 shows the shear rate dependence of the intrinsic shear | 3,068,603 | 11722037 | 0 | 16 |
viscosity [44] [η * ] = lim x1→0 (η * − η * s )/η * s for the mass ratio m 1 /m 2 = 0.1 (light impurity) and different values of the (common) coefficient of restitution. We observe that the intrinsic viscosity [η * ] is clearly different from zero for shear rates larger than its corresponding critical value. On the other hand, its magnitude is smaller than the one obtained for the order parameter E 1 /E (see Fig. 7) V. DISCUSSION AND CONCLUSION In this paper we have analyzed the dynamics of an impurity immersed in a granular gas subject to USF. The study has been performed in two successive steps. First, the pressure tensor of a granular binary mixture of inelastic Maxwell gases under USF has been obtained from an exact solution of the Boltzmann equation. This solution applies for arbitrary values of the shear rate a and the parameters of the mixture, namely, the mole fraction x 1 = n 1 /n, the mass ratio µ ≡ m 1 /m 2 and the coefficients of restitution α 11 , α 22 and α 12 . Then, the tracer limit (x 1 → 0) of the above solution has been carefully considered, showing that the relative contribution of the tracer species to the total properties of the mixture does not necessarily vanish as x 1 → 0. This surprising result extends to inelastic gases some results derived some time ago for ordinary gases [23]. The above phenomenon can be seen as | 3,068,604 | 11722037 | 0 | 16 |
a non-equilibrium phase transition, where the relative contribution of the impurity to the total kinetic energy E 1 /E plays the role of an order parameter [45]. The transition problem addressed here has been analyzed in the framework of the Boltzmann equation with Maxwell kernel. The key advantage of inelastic Maxwell models, in comparison with the more realistic inelastic hard sphere model, is that the collisional moments of the Boltzmann collision operators J rs [f r , f s ] can be exactly evaluated in terms of the velocity moments of f r and f s , without the explicit knowledge of these velocity distribution functions. Here, we have explicitly determined the collisional moments associated with the second-degree velocity moments to get the pressure tensor of the mixture. In addition, the collision rates ω rs appearing in the operators J rs [f r , f s ] have been chosen to be time independent so that the interaction model allows to disentangle the effects of collisional dissipation (accounted for by the coefficients of restitution) from those of boundary conditions (embodied in the reduced shear rate a * defined as a * = a/ν 0 , ν 0 being a characteristic collision frequency). Consequently, within this model, collisional dissipation and viscous heating generally do not compensate, so that the granular temperature increases (decreases) with time if viscous heating is larger (smaller) than collisional cooling. In our system, the temperature T 2 of the gas particles (T 2 ≃ T in the disordered phase when x 1 → | 3,068,605 | 11722037 | 0 | 16 |
0) changes in time due to two competing effects: the viscous heating term (−aP xy ) and the inelastic cooling term (ζT 2 ). In fact, for long times, xy (here, ζ * = ζ/ν 0 and P * xy = P xy /p). Since P * xy < 0, the cooling rate ζ * can be interpreted as the "thermostat" parameter needed to get a stationary value for the temperature T 2 of the gas particles. At a given value of the shear rate, the cooling rate increases with dissipation and so, T 2 decreases as α rs decreases. The tracer particles are also subject to two antagonistic mechanisms. On the one hand, T 1 → ∞ due to viscous heating and on the other hand, collisions with the gas particles tend to "thermalize" T 1 to T 2 . Both effects are accounted for by the root λ 2 , the temperature ratio T 1 /T 2 grows without bounds. The parameter ranges where such a requirement is met define the ordered "pockets" of the phase diagram, and where the energy ratio E 1 /E -explicitly worked out here-reaches a finite value. We have found that two different families of ordered phase can be encountered • a light impurity phase, provided that the mass ratio µ does not exceed the threshold µ (−) th given by Eq. (51). Such a phase always exists for shear rates larger than a certain critical value, but can also be observed at vanishing shear in cases of asymmetric | 3,068,606 | 11722037 | 0 | 16 |
collisional dissipation, whenever gas-gas collisions are sufficiently more dissipative that intruder-gas collisions (α 12 > (1 + α 2 22 )/2). • a heavy impurity phase, that -unlike the light impurity phase-cannot accommodate large shear, and requires a * < a * (+) , where the threshold a * (+) is given by Eq. (53). The fact that a * and α rs are independent parameters [unless a * takes the specific value a * s given by the steady state condition (24)] allows one to carry out a clean analytical study of the combined effect of both control parameters on the properties of the impurity particle. It is however important to bring to the fore the precise coupling between shear and collisional dissipation that the steady state condition Eq. (24) implies. The answer depends on the phase considered, ordered or disordered and the energy balance embodied in Eq (24) can be expediently expressed as max(λ where we have introduced the notation β 12 ≡ µ 21 (1 + α 12 ). Finally, the steady state condition implies a * = a * (+) (resp. a * = a * 1,s ) in the disordered (resp ordered) case. The corresponding line in the shear versus mass ratio plane is shown in Fig. 11, for a given set of dissipation parameters that corresponds to one of the situations analyzed in Fig. 6. It appears that remaining on the steady state line shown by the thick curve, one spans the three possible regimes: light tracer ordering at small | 3,068,607 | 11722037 | 0 | 16 |
µ, disordered phase at intermediate values, and heavy tracer order at larger µ. We conclude here that the scenario uncovered in our analysis is not an artifact of having decoupled shear from dissipation, and we emphasize that from a practical point of view, studying the transient regime (before the steady state occurs) anyway offers the possibility to enforce the above decoupling. In addition, we would like to stress here that, in spite of the approximate nature of our plain vanilla Maxwell model, the results obtained for binary mixtures under steady USF [13] compare quite well with Monte Carlo simulations of inelastic hard spheres [19]. This can be seen in Fig. 12 for the (reduced) elements of the pressure tensor P * ij in the steady shear flow state defined by the condition (24). Therefore, we expect that the transition found in this paper is not artefactual and can be detected in the case of hard spheres interaction. It is quite natural -when analyzing the dynamics of an impurity immersed in a background of mechanically different particles-to invoke two assumptions. First, that the state of the solvent (excess component 2) is not affected by the presence of the tracer (solute) particles 1. Second, that the effect on the state of the solute due to collisions among the tracer particles themselves can be neglected. We have seen that the second expectation is correct (the coefficient of restitution α 11 is immaterial in the tracer limit, a property that is not obvious from the cumbersome analytical formulas reported here), | 3,068,608 | 11722037 | 0 | 16 |
but the first expectation is invalidated in the regions where the ordered phase sets in. Consequently, the seemingly natural "Boltzmann-Lorentz" point of view -with the oneparticle velocity distribution function f 2 of the granular gas obeying a (closed) nonlinear Boltzmann kinetic equation while the one-particle velocity distribution function f 1 of the impurity particle obeys a linear Boltzmann-Lorentz kinetic equation-breaks down. Our results show that collisions of type 2-1 affect f 2 , despite being much less frequent than collisions of type 2-2. We conclude here that, rather unexpectedly, the tracer problem is as complex as the general case of a binary mixture at arbitrary mole fractions. Ref. [19]). The parameters are x1 = 0.5 (equimolar mixture) and µ = 2. It should be noted that for a meaningful comparison, a * and α need to be coupled (see Eq. (24)), since inelastic hard spheres enjoy a steady state for a precise value of the reduced shear rate, that depends on the coefficient of restitution α. ACKNOWLEDGMENTS The research of V. G. has been supported by the Ministerio de Educación y Ciencia (Spain) through Grant No. FIS2010-16587, partially financed by FEDER funds and by the Junta de Extremadura (Spain) through Grant No. GR10158. Appendix A: Energy ratio p * 1 and shear stress P * xy The expression of the energy ratio p * 1 can be obtained from Eq. (31). It can be written as where Once the energy ratio is known, the remaining relevant elements of the pressure tensor can be easily obtained from | 3,068,609 | 11722037 | 0 | 16 |
Eqs. (31)- (34). In particular, the shear stress P * xy is given by P * xy = P * 1,xy + P * 2,xy where (A7) Appendix B: Derivation of λ When x 1 → 0, the sixth-degree equation (30) for the rates λ factorizes into two cubic equations given by rs denote the zeroth-order contributions to the expansion of A * rs ≡ A rs /ν 0 and B * rs ≡ B rs /ν 0 , respectively, in powers of x 1 . They are given by (B6) Equation (B1) is associated with the time evolution of the excess component. Its largest root is given by Eq. (36). On the other hand, Eq. (B2) gives the transient behavior of the impurity. Its largest root is given by Eq. (39). In this Appendix, we provide some of the expressions used along the text in the tracer limit. First, the quantities D, ∆ 0 and ∆ 1 appearing in Eq. (40) are given by (C3) In these equations, A i (a) can be written as where i a * 4 + 2dX (2) i a * 2 + d 2 (B With respect to the shear stress P * xy , in the disordered phase (λ (1 − p * 1 ) , where p * 1 is given by Eq. (44). Consequently, according to Eq. (55), the non-linear shear viscosity η * in the disordered phase is while in the ordered phase the result is [2] A. Barrat ture, the energy ratio E1/E plays | 3,068,610 | 11722037 | 0 | 16 |
the role of the magnetization, and the mole fraction x1 plays the role of the external magnetic field. One can then wonder what is the symmetry broken for our system, to which the order parameter would be associated. Given that near equilibrium E1/E ∝ x1, then x −1 1 E1/E is invariant under the change x1 → βx1, where β is an arbitrary constant. This invariance is broken in the ordered phase since the mean kinetic energy of the impurity is very much larger than the one corresponding to the excess particles. | 3,068,611 | 11722037 | 0 | 16 |
Open and Unoriented Strings from Topological Membrane - I. Prolegomena We study open and unoriented strings in a Topological Membrane (TM) theory through orbifolds of the bulk 3D space. This is achieved by gauging discrete symmetries of the theory. Open and unoriented strings can be obtained from all possible realizations of $C$, $P$ and $T$ symmetries. The important role of $C$ symmetry to distinguish between Dirichlet and Neumman boundary conditions is discussed in detail. Introduction Although originally (and historically) open string theories were considered as theories by themselves, it soon become evident that, whenever they are present, they come along with closed (non-chiral) strings. Moreover open string theories are obtained from closed string theories by gauging certain symmetries of the closed theory (see [1] and references therein for a discussion of this topic). The way to get open strings from closed strings is by gauging the world-sheet parity [1][2][3], Ω : z → −z. That is we impose the identification σ 2 ∼ = −σ 2 , where z = σ 1 + iσ 2 andz = σ 1 − iσ 2 ) is the complex structure of the world-sheet manifold. The spaces obtained in this way can be of two types: closed unoriented and open oriented (and unoriented as well). These last ones are generally called orbifolds and the singular points of the construction become boundaries. The states (operators and fields of the theory in general) of the open/unoriented theory are obtained from the closed oriented theory by projecting out the ones which have negative | 3,068,612 | 119075145 | 0 | 16 |
eigenvalues of the parity operator. This is obtained by building a suitable projection operator (1 + Ω)/2 such that only the states of positive eigenvalues are kept in the theory. Namely the identification X I (z,z) ∼ = X I (z, z) or X I L (z) ∼ = X I R (z) (in terms of the holomorphic and antiholomorphic parts of X = X L + X R ) holds. Another construction in string theory is orbifolding the target space of the theory under an involution of some symmetry of that space. In this work we are going to consider only a Z 2 involution, imposing the identification X I ∼ = −X I , where X I are the target space coordinates. When combining both constructions, world-sheet and target space orbifolding, we obtain open/unoriented theories in orbifolds [4][5][6][7] or orientifolds (X I (z,z) = −X I (z, z)), implying the existence of twisted sectors in the open/unoriented theories. Further to the previous discussion both sectors (twisted and untwisted) need to be present for each surface in order to ensure modular invariance of the full partition function [1,8,18]. One point we want to stress is that twisting in open strings can, for the case of a Z 2 target space orbifold, be simply interpreted as the choice of boundary conditions: Neumman or Dirichlet. Toroidal compactification is an important construction in string theories and in the web of target space dualities. Early works considered also open string constructions in these toroidal backgrounds [8,9]. In these cases | 3,068,613 | 119075145 | 0 | 16 |
we have some compactified target space coordinates, say X J (z+2πi,z−2πi) ∼ = X J (z,z)+2πR (R is the radius of compactification of X J ), the twisted states in the theory are the ones corresponding to the points identified under X J (z + 2πi,z − 2πi) ∼ = −X J (z,z) + 2πR or in terms of the holomorphic and antiholomorphic parts of X this simply reads X I L (z) ∼ = −X I R (z). An important result coming from these constructions is that the gauge group of the open theory, the Chan-Paton degrees of freedom carried by the target space photon Wilson lines (only present in open theories) are constrained, both due to dualities of open string theory [8] and to modular invariance of open and unoriented theories [8][9][10][11]. This will result in the choice of the correct gauge group that cancels the anomalies in the theory. One fundamental ingredient of string theory is modular invariance. Although for bosonic string theory the constraints coming from genus 1 amplitudes are enough to ensure modular invariance at generic genus g, it becomes clear that once the fermionic sector of superstring theory is considered it is necessary to consider genus 2 amplitude constraints. For closed strings (types II and 0) the modular group at genus g is SP L(2g, Z) and the constraints imposed by modular invariance at g = 2 induce several possible projections in the state space of the theory [12][13][14][15][16] such that the resulting string theories are consistent. Among them are | 3,068,614 | 119075145 | 0 | 16 |
the well known GSO projections [17] that insure the correct spin-statistics connection, project out the tachyon and ensure a supersymmetric effective theory in the 10D target space. Once we consider an open superstring theory (type I) created by orbifolding the world-sheet parities, for each open (and/or unoriented) surface a Relative Modular Group still survives the orbifold at each genus g [18]. Again in a similarly way to the closed theory the modular invariance under these groups will result in generalized GSO projections [18][19][20][21]. For a more recent overview of the previous topics see [22,23] (see also [24] for an extensive explanation of them). Closed string theories are obtained as the effective boundary theory, their worldsheet is the closed boundary ∂M. Obtaining open string theory raises a problem, we need a open world-sheet to define them. But the boundary of a boundary is zero, ∂∂M = 0. So naively it seems that TM cannot describe open strings since world-sheets are already a boundary of a 3D manifold. The way out is to consider orbifolding of the bulk theory. In this way the fixed points of the orbifold play the role of the boundary of the 2D boundary of the 3D membrane. This proposal was first introduced by Horava [47] in the context of pure Chern-Simons theories. We are going to extend his results to TMGT and reinterpret the orbifolded group as symmetries of the full gauge theory. Other works have developed Horava's idea. For a recent study on WZNW orbifold constructions see [48] (and references therein) For | 3,068,615 | 119075145 | 0 | 16 |
an extensive study, although in a more formal way than our work, of generic Rational Conformal Field Theories (RCFT) with boundaries from pure 3D Chern-Simons theory see [49] (and references therein). Nevertheless previously the monopole processes were not studied. These are crucial for describing the winding modes and T-duality in compact RCFT from the TM point of view and, therefore, in compactified string theories. We consider an orbifold of TM(GT) such that one new boundary is created at the orbifold fixed point. To do this we gauge the discrete symmetries of the 3D theory, namely P T and P CT . Several P 's are going to be defined as generalized parity operations. C and T are the usual 3D QFT charge conjugation and time inversion operations (see [50] for a review). The orbifolding of the string target space corresponds in pure Chern-Simons membrane theory to the quotient of the gauge group by a Z 2 symmetry [45]. As will be shown, in the full TM(GT), the discrete symmetry which will be crucial in this construction is charge conjugation C. Besides selecting between twisted and untwisted sectors in closed unoriented string theory it will also be responsible for setting Neumann and Dirichlet boundary conditions in open string theory. In this work we are not going to consider more generic orbifold groups. There are two main new ideas introduced in this work. Firstly the use of all possible realizations of P , C and T combinations, which constitute discrete symmetries of the theory, as the orbifold group. | 3,068,616 | 119075145 | 0 | 16 |
Although the mechanism is similar to the one previously studied by Horava for pure Chern-Simons theory, the presence of the Maxwell term constrains the possible symmetries to P T and P CT type only. Also the interpretation of the orbifold group as the discrete symmetries in the quantum theory is new, as is the interpretation of charge conjugation C which selects between Neumman and Dirichlet boundary conditions. This symmetry explains the T-duality of open strings in the TM framework. It is a symmetry of the 3D bulk which exchanges trivial topological configurations (without monopoles) with non-trivial topological configurations (with monopoles). In terms of the effective boundary CFT (string theory) this means exchanging Kaluza-Klein modes (no monopole effects in the bulk) with winding number (monopole effects in the bulk). In section 2 we start by introducing genus 0 (the sphere), and genus 1 (the torus), Riemann surfaces and their possible orbifolds under discrete symmetries which we identify with generalized parities P . Section 3 gives an account of Neumann and Dirichlet boundary conditions in usual CFT using the Cardy method [51] of relating n point full correlation functions in boundary Conformal Field Theory with 2n chiral correlation functions in the theory without boundaries. Then, in section 4 we give a brief overview of the discrete symmetries of 3D QFT and use it to orbifold TM(GT). We enumerate the 3D configurations compatible with the several orbifolds, both at the level of the field configurations and of the particular charge spectrums corresponding to the resulting theories. It naturally emerges | 3,068,617 | 119075145 | 0 | 16 |
from the 3D membrane that the configurations compatible with P CT correspond to Neumann boundary conditions (for open strings) and to untwisted sectors (for closed unoriented). The configurations compatible with P T correspond to Dirichlet boundary conditions (for open strings) and twisted sectors (for closed unoriented). The genus 2 constraints are discussed here although a more detailed treatment is postponed for future work. Further it is shown that Neumann (untwisted) corresponds to the absence of monopole induced processes while for Dirichlet (twisted) these processes play a fundamental role. A short discussion on T-duality show that it has the same bulk meaning as modular invariance, they both exchange P T ↔ P CT . Riemann Surfaces: from Closed Oriented to Open and Unoriented Any open or unoriented manifold Σ u can, in general, be obtained from some closed orientable manifolds Σ under identification of a Z 2 (or at most two Z 2 ) involution such that each point in Σ u has exactly two corresponding points in Σ conjugate in relation to the Z 2 involution(s). The pair (x, −x) in the last equation is symbolic, the second element stands for the action of the group Z 2 , z 2 (x) = −x, in the manifold. Usually this operation is closely related with parity as will be explained bellow. Although in this work our perspective is that we start from a full closed oriented theory and orbifold it, there is the reverse way of explaining things. This means that any theory defined in an open/unoriented | 3,068,618 | 119075145 | 0 | 16 |
manifold is equivalently defined in the closed/oriented manifold which doubles (consisting of two copies of) the original open/unoriented. Let us summarize how to obtain the disk D 2 (open orientable) and projective plane RP 2 (closed unorientable) out of the sphere S 2 and the annulus C 2 (open orientable), the Möbius Strip (open unorientable) and Klein bottle K 2 (closed unorientable) out of the torus T 2 . The Projective Plane and the Disk obtained from the Sphere For simplicity we choose to work in complex stereographic coordinates (z = x 1 + ix 2 , z = x 1 − ix 2 ) such that the sphere is identified with the full complex plane. The sphere has no moduli and the Conformal Killing Group (CKG) is P SL(2, C). A generic element of this group is (a, b, c, d) with the restriction ad − bc = 1. It acts in a point as It has then six real parameters, that is, six generators. That is to say that the sphere has six Conformal Killing Vectors (CKV's). It is necessary to use two coordinate charts to cover the full sphere, one including the north pole and the other one including the south pole. Usually it is enough to analyze the theory defined on the sphere only for one of the patches but it is necessary to check that the transformation between the two charts is well defined. In stereographic complex coordinates the map between the two charts (with coordinates z,z and u,ū) is given | 3,068,619 | 119075145 | 0 | 16 |
by z → 1/u andz → 1/ū. The disk D 2 can be obtained from the sphere under the identification This result is graphically pictured in figure 1 and consists in the involution of the manifold S 2 by the group Z P 1 , D 2 = S 2 /Z P 1 . There are one boundary corresponding to the real line in the complex plane and the disk is identified with the upper half complex plane. It is straightforward to see that the non trivial element of Z P 1 is nothing else than the usual 2D parity transformation The CKG of the disk is the subgroup of P SL(2, C) which maintains constraint (2.3), that is P SL(2, R). From the point of view of the fields defined in the sphere this corresponds to the usual 2D parity transformation. In order that the theory be well defined in the orbifolded sphere we have to demand the fields of the theory to be compatible with the construction where the first equation applies to scalar fields and the second to vectorial ones. For tensors of generic dimensions d (e.g. the metric or the antisymmetric tensor) the transformation is easily generalized to be T (x) = P d 1 T (P 1 (x)). In order to orbifold the theory defined on the sphere we can introduce the projection operator which projects out every operator with odd parity eigenvalue and keeps in the theory only field configurations compatible with the Z 2 involution. To obtain the projective | 3,068,620 | 119075145 | 0 | 16 |
plane RP 2 we need to make the identification This result is graphically pictured in figure 2 and again is an involution of the sphere RP 2 = S 2 /Z P 2 2 . The resulting space has no boundary and no singular points. But it is now an unoriented manifold. This identification can be thought of as two operations. The action of the element α = (0, −1, 1, 0) ∈ Z α 2 ⊂ SL(2, C) followed by the operation of parity as given by (2.4). Note that α(z) = −1/z but P 1 α(z) = −1/z as desired. In this case we can define a new parity operation P 2 ∈ Z P 2 2 = Z P 1 2 × Z α 2 as From the point of view of the fields defined in the sphere we could use the usual parity transformation since any theory defined in the sphere should be already invariant under transformation (2.2) such that P SL(2, C) is a symmetry of the theory. But in order to have a more transparent picture we use the definition (2.8) of P 2 and demand that where the first equation concerns to scalar fields and the second to vectorial ones. For tensors of generic dimensions d (as the metric or the antisymmetric tensor) the transformation is again easily generalized to be T (x) = P d 2 T (P 2 (x)). The CKG is now SO(3), the usual rotation group. It is the subgroup of P SL(2, C)/Z α | 3,068,621 | 119075145 | 0 | 16 |
2 that maintains constraint (2.3) The annulus, Möbius strip and Klein bottle from the Torus Let us proceed to genus one closed orientable manifold, the torus. It is obtained from the complex plane under the identifications There are two modular parameters τ = τ 1 + iτ 2 and two CKV's. The action of the CKG, the translation group in the complex plane, is with a and b real. The metric is simply |dx 1 + τ dx 2 | and the identifications on the complex plane are invariant under the two operations These operations constitute the modular group P SL(2, Z). That is with a, b, c, d ∈ Z and ad − bc = 1. The annulus C 2 (or topologically equivalent, the cylinder) is obtained from the torus with τ = iτ 2 under the identification This result is symbolically picture in figure 3. There is now one modular parameter τ 2 and no modular group. There is only one CKV being the CKG action given by z ′ = z + ib, translation in the imaginary direction. In terms of the fields defined in the torus this correspond to the projection under the parity operation Ω : z → −z z → −z (2.15) The Möbius strip M 2 can be obtained from the annulus (obtained from the torus with τ = 2iτ 2 ) by the identification under the elementã [24] of the translation groupã Note thatã belongs to the translation group of the torus, not of the disk, and that | 3,068,622 | 119075145 | 0 | 16 |
a 2 = 1. This construction corresponds to two involutions, so the orbifolding group is constituted by two Z 2 's, where ⊂ × stands for the semidirect product of groups. Thus the ratio of areas between the Möbius strip and the original torus is 1/4 contrary to the 1/2 of the remaining open/unoriented surfaces obtained from the torus, due to the extra projection operator (1+ã)/2 taking from the annulus to the strip. In terms of the fields living on the torus we can think of this identification as the projection under a new discrete symmetry, which we also call paritỹ Although this operation does not seem to be a conventional parity operation note that, applying it twice to some point, we retrieve the same point,Ω 2 = 1. It is in this sense a generalized parity operation. The previous construction is presented, for example, in Polchinski's book [24]. Let us note however that one can build the Möbius strip directly from a torus [1] with moduli τ = 1/2 + iτ 2 under the involution by Ω as given in (2.15) 3 . In this case the ratio of areas between the original torus and the involuted surface is 1/2 as the other involutions studied in this section. As we will show later both constructions correspond to the same region on the complex plane. The first one results from two involutions of a torus (τ = 2iτ ) with double the area of the second construction (τ = iτ ). In this sense both constructions | 3,068,623 | 119075145 | 0 | 16 |
are equivalent. The Möbius strip orbifolding is pictured in figure 4. Again there is one modular parameter τ 2 and no modular group. The only CKV is again the translation in the imaginary direction. The Klein bottle K 2 is obtained from the torus with τ = 2iτ 2 under the identification This result is pictured in figure 5. for Ω and τ = 2iτ 2 forΩ and Ω ′ . Note that M 2 can also be obtained from the torus with τ = 1/2 + iτ 2 considering the parity Ω. In the labels of the last line the first letter stands for Open or Close surface while the second letter stands for Oriented or Unoriented. The bottle is the involution of the torus K 2 = T 2 /Z Ω ′ 2 , has one parameter CKG with one CKV, translations in the imaginary direction. There is one modulus τ 2 and no modular group. The resulting manifold has no boundary and no singular points but is unoriented. Again we can define a new parity transformation Ω ′ We summarize in table 1 all the parity operations we have just studied together with the resulting involutions (or orbifolds). Conformal Field Theory -Correlation Functions and Boundary Conditions To study string theory we need to know the world-sheet CFT. In a closed string theory they are given by CFT on a closed Riemann surface, the simplest of them is the sphere, or equivalently the complex plane. To study open strings we need to study CFT | 3,068,624 | 119075145 | 0 | 16 |
on open surfaces. As was shown by Cardy [51] n-point correlation functions on a surface with a boundary are in one-to-one correspondence with chiral 2n point correlation functions on the double surface 4 (for more details and references see [52]). We will study the disk and the annulus, so we double the number of charges (vertex operators) by inserting charges ±q (vertex operators with ∆ = 2q 2 /k) in the Parity conjugate points. Note that the sign of the charges inserted depends on the type of boundary conditions that we want to impose but the conformal dimension of the corresponding vertex operator is the same. We summarize the 2, 3 and 4-point holomorphic correlation functions of vertex operators for the free boson where in all the cases q i = 0, otherwise they vanish. Disk We will take the disk as the upper half complex plane. As explained before it is obtained from the sphere (the full complex plane) by identifying each point in the lower half complex plane with it's conjugate in the upper half complex plane. In terms of correlation functions where we replaced z = x + iy in the the first equation of (3.1), y is the distance to the real axis while x is taken to be the horizontal distance (parallel to the real axis) between vertex insertions. Dirichlet Boundary Conditions As it is going to be shown, when the mirror charge have opposite sign the boundary conditions are Dirichlet. The 2-point correlation function restricted to the upper half plane | 3,068,625 | 119075145 | 0 | 16 |
is simply the expectation value Insertion of vertex operators (from the unity) in the boundary is not compatible with the boundary conditions since the only charge that can exist there is q = 0 (since q = −q = 0 in the boundary). Taking the limit y → 0 the expectation value (3.3) blows up but this should not worry us, near the boundary the two charges annihilate each other. This phenomena is nothing else than the physical counterpart of the operator fusion rules 3-point correlation functions cannot be used for the same reason, one of the insertions would need to lie in the boundary but that would mean q 3 = 0, the other two charges had to be inserted symmetrically in relation to the real axis and would imply q 1 = −q 2 . This reduces the 3-point correlator to a 2-point one in the full plane. For 4-point vertex insertions consider q 1 and q 3 in the upper half plane, q 2 (inserted symmetrically to q 1 ) and q 4 (inserted symmetrically to q 3 ) in the lower half plane. As pictured in figure 6 the most generic configurations is q 1 = −q 2 = q and q 3 = −q 4 = q ′ . Making z 2 =z 1 = −iy and z 4 =z 3 = x − iy ′ we obtain the corresponding 2-point correlators in the upper half plane Again note that we cannot insert boundary operators without changing the boundary conditions. In | 3,068,626 | 119075145 | 0 | 16 |
the limit x → ∞ both correlators behave like When we approach the boundary the correlators go to infinite independently of the value of x. This fact can be explained by the kind of boundary conditions we are considering, they are such that when the fields approach the boundary they become infinitely correlated independently of how far they are from each other. Therefore this must be Dirichlet boundary conditions, the fields are fixed along the boundary, furthermore, as stated before their expectation value is 1 . It doesn't mater how much apart they are, they are always correlated on the boundary. The tangential derivative to the boundary of the expectation value ∂ x φ | ∂D 2 = 0 also agrees with Dirichlet boundary conditions. Neumann Boundary Conditions For the case of the mirror charge having the same sign of the original one the boundary conditions will be Neumann. The expectation value for the fields in the bulk vanishes since the 2-point function φ q (z 1 )φ q (z 2 ) = 0 in the full plane. Nevertheless we can evaluate directly the non-zero 2-point correlation function in the boundary Note that contrary to the previous discussion, concerning Dirichlet boundary conditions, in this case q = 0 on the boundary since the mirror charges have the same sign and the correlation function vanishes in the limit x → ∞ indicating that the boundary fields become uncorrelated. The 3-point correlation function in the full plane must be considered with one charge −2q in the boundary and | 3,068,627 | 119075145 | 0 | 16 |
two other charges q inserted symmetrically in relation to the real axis (see figure 6). In the upper half plane this corresponds to one charge insertion in the boundary and one in the bulk Note that in the limit y → 0 the fusion rules apply and we obtain (3.6) with ∆ replaced by 4∆. For the 2-point function in the upper plane we have to consider the 4-point correlation function in the full plane with q 1 = q 2 = −q 3 = −q 4 = q, where q 2 is inserted symmetrically to q 1 in relation to the real axis and q 4 to q 3 . We obtain the bulk correlator Again in the limit x → ∞ this correlator vanishes. This corresponds to Neumann boundary conditions. The normal derivative to the boundary of (3.8) vanishes on the boundary ∂ y φ(0)φ(x) | ∂D 2 = 0. For the case of one compactified free boson the process follows in quite a similar way. The main difference resides in the fact that the right and left spectrum charges are different. Taking a charge q = m + kn/4 its image charge is now ±q, wherē q = m − kn/4. In this way we have to truncate the spectrum holding q = −q = kn/4 for Dirichlet boundary conditions, and q =q = m for Neumann boundary conditions, in a pretty similar way as it happens in the Topological Membrane. We summarize in figure 6 the results derived here. Annulus We consider | 3,068,628 | 119075145 | 0 | 16 |
the annulus to be a half torus. For simplicity we take the torus to be the region of the complex plane [−π, π] × [0, 2πτ ] (and the annulus the region [0, π] × [0, 2πτ ]). We use z = x + iy with x ∈ [−π, π] and y ∈ [0, 2πτ ]. Here y is the vertical distance (parallel to the imaginary axis) between vertex insertions while x is taken to be the distance to the imaginary axis. Dirichlet Boundary Conditions Considering mirror charges with opposite sign, 2-point correlations in the torus correspond to the bulk expectation value in the annulus As in the case of the disk, it blows up in the boundary. But in the boundary this correlation function is not valid since the two charges annihilate each other. Therefore the only possible charge insertions in the boundary are q = 0, that is the identity operator. Again 3-point correlation functions cannot be used in this case. For 4-point vertex insertion consider q 1 and q 3 inserted to the right of the imaginary axis and q 2 and q 4 their mirror charges. The most generic configuration is We obtain the 2-point correlation function in the annulus Again the same arguments used for the disk apply. There cannot exist boundary insertions other than the identity and the tangential derivative to the boundary ∂ y φ | ∂C 2 = 0 vanish. Neumann Boundary Conditions Considering now the mirror charges having the same sign, again the fields in the bulk | 3,068,629 | 119075145 | 0 | 16 |
have zero expectation value. But the 2-point boundary correlation function is computed to be where we take one insertion in each boundary. In the case that the insertions are in the same boundary the factor of π 2∆ is absent. The 3-point function in the torus corresponds either to 2-point function in the annulus (taking only one insertion in the boundary) or to 3-point function (taking all the insertions in the boundaries). Taking one insertion in the bulk φ q (x, 0) (with mirror image φ q (−x, 0)) and other in the boundary φ −2q (π, y) we obtain If the insertion is the boundary x = 0 the factor of π 2 is absent. As an example of two insertions in the boundaries take them to be both in the boundary x = 0, we obtain x 2 + (y + y ′ ) 2 y 2 (x 2 + y ′2 ) 2qq ′ /k (3.14) We can stop here, for our purposes it is not necessary to exhaustively enumerate all the possible cases. As expected the normal derivative to the boundary of these correlation functions (∂ x . . . ) vanishes at the boundary. These results are summarized in figure 7. For the case of one compactified free boson the process follows as explained before. The spectrum must be truncated holding q = −q = kn/4 for Dirichlet boundary conditions and q =q = m for Neumann boundary conditions. TM(GT) Is now time to turn to the 3D TM(GT). In this | 3,068,630 | 119075145 | 0 | 16 |
section we present results derived directly from the bulk theory and its properties. The derivations of the results presented here are in agreement with the CFT arguments in the last section. Take for the moment a single compact U(1) TMGT corresponding to c = 1 CFT with action where M = Σ × [0, 1] has two boundaries Σ 0 and Σ 1 . Σ is taken to be a compact manifold, t is in the interval [0, 1] and (z,z) stand for complex coordinates on Σ. From now on we will use them by default. As widely known this theory induces new degrees of freedom in the boundaries, which are fields belonging to 2D chiral CFT's theories living on Σ 0 and Σ 1 . The electric and magnetic fields are defined as and the Gauss law is simply Upon quantization the charge spectrum is for some integers m and n. Furthermore it has been proven in [32,39] that, for compact gauge groups and under the correct relative boundary conditions, one insertion of Q on one boundary (corresponding to a vertex operator insertion on the boundary CFT) will, necessarily, demand an insertion of the chargē on the other boundary. We are assuming this fact through the rest of this paper. Our aim is to orbifold TM theory in a similar way to Horava [47], who obtained open boundary world-sheets through this construction. We are going to take a path integral approach and reinterpret it in terms of discrete P T and P CT symmetries of | 3,068,631 | 119075145 | 0 | 16 |
the bulk 3D TM(GT). Horava Approach to Open World-Sheets Obtaining open string theories out of 3D (topological) gauge theories means building a theory in a manifold which has boundaries (the 2D open string world-sheet) that is already a boundary (of the 3D manifold). This construction raises a problem since the boundary of a boundary is necessarily a null space. One interesting way out of this dilemma is to orbifold the 3D theory, then its singular points work as the boundary of the 2D boundary. Horava [47] introduced an orbifold group G that combines the world-sheet parity symmetry group Z W S 2 (2D) with two elements {1, Ω}, together with a target symmetryG of the 3D theory fields With this construction we can get three different kind of constructions. Elements of the kind h =h×1 Z W S 2 induce twists in the target space (not acting in the world-sheet at all), for elements ω = 1G × Ω we orbifold the world-sheet manifold (getting an open world-sheet) without touching in the target space and for elements g 1 =g 1 × Ω we obtain exotic world-sheet orbifold. In this last case it is further necessary to have an element corresponding to the twist in the opposite direction g 2 =g 2 × Ω. To specify these twists on some world-sheet it is necessary to define the monodromies of fields on it. Taking the open string C o = C/Z 2 as the orbifold of the closed string C π(C o ) = D ≡ Z | 3,068,632 | 119075145 | 0 | 16 |
2 * Z 2 ≡ Z 2 ⊂ ×Z (4.7) * being the free product and ⊂ × the semidirect product of groups. D is the infinite dihedral group, the open string first homotopy group. So the monodromies of fields in C o corresponds to a representation of this group in the orbifold group, Z 2 * Z 2 → G, such that the commutative triangle is complete. The partition function contains the sum over all possible monodromies where τ is the moduli of the manifold. The monodromies g 1 , g 2 and h are elements of G as previously defined satisfying g 2 i = 1 and [g i , h] = 1. It will be shown that P CT plays the role of one of such symmetries with g 1 = g 2 . It is in this sense one of the most simple cases of exotic world-sheet orbifolds. The string amplitudes can be computed in two different pictures. The loopchannel corresponds to loops with length τ of closed and open strings and the amplitudes are computed as traces over the Hilbert space. The tree-channel corresponds to a cylinder of lengthτ created from and annihilated to the vacua through boundary (|B ) and/or crosscaps (|C ) states. Comparing both ways for the same amplitudes we obtain where I acts in t as Time Inversion t → 1 − t. This construction is presented in figure 8. In terms of the action and fields in the theory Horava used the same approach of extending | 3,068,633 | 119075145 | 0 | 16 |
them to the doubled manifold Since there is a one-to-one correspondence between the quantum states of the gauge theory on M and the blocks of the WZNW model, we may write where Ψ i stands for a basis of the Hilbert space H Σ . The open string counterpart in the orbifolded theory is Z Σo = a i Ψ i ∈ H Σ (4.14) which also agrees with the fact that in open CFT's the partition function is the sum of characters (instead of the sum of squares) due to the holomorphic and antiholomorphic sectors not being independent. Discrete Symmetries and Orbifold of TM(GT) Following the discussion of section 2 and section 4.1, it becomes obvious that the parity operation plays a fundamental role in obtaining open and/or non-orientable manifolds out of closed orientable ones. Hence obtaining open/unorientable theories out of closed orientable theories. Generally there are several ways of defining parity. The ones we are interested in have already been presented here. For the usual ones, P 1 and Ω defined in (2.4) and (2.15), the fields of our 3D theory transform like where Λ is the gauge parameter entering into U(1) gauge transformations. Under these two transformations the action transforms as The theory is clearly not parity invariant. Let us then look for further discrete symmetries which we may combine with parity in order to make the action (theory) invariant. Introduce time-inversion, T : t → 1 − t, implemented in this non-standard way due to the compactness of time. Note that t | 3,068,634 | 119075145 | 0 | 16 |
= 1/2 is a fixed point of this operation. Upon identification of the boundaries as described in [39] the boundary becomes a fixed point as well. It remains to define how the fields of the theory change under this symmetry. There are two possible transformations compatible with gauge transformations, A Λ (t, z,z) = A(t, z,z) + ∂Λ(t, z,z). They are: and where we defined C, charge conjugation, as A µ → −A µ . This symmetry inverts the sign of the charge, Q → −Q, as usual. These discrete symmetries together with parity P or Ω are the common ones used in 3D Quantum Field Theory. When referring to parity in generic terms we will use the letter P . Under any of the T and CT symmetries the action changes in the same fashion it does for parity P , as given by (4.16). In this way any of the combinations P T and P CT are symmetries of the action, S → S. Gauging them is a promising approach to define the TM(GT) orbifolding. It is now clear why we need extra symmetries, besides parity, in order to have combinations of them under which the theory (action) is invariant. In general, whatever parity definition we use, these results imply that P T and P CT are indeed symmetries of the theory. We can conclude straight away that any of the two previous symmetries exchange physically two boundaries working as a mirror transformation with fixed point (t = 1/2, z =z = x) (corresponds | 3,068,635 | 119075145 | 0 | 16 |
actually to a line) as pictured in figure 9. We are considering that, whenever there is a charge insertion in one boundary of q=m+kn/4, it will exist an insertion ofq = m − kn/4 in the other boundary [30,39]. Under the symmetries P T and P CT as given by (4.15), (4.17) and (4.18) the boundaries will be exchanged as presented in figure 9. In the case of P CT the charges will simply be swaped but in the case of P T their sign will be change q → −q. Note that Σ1 2 = Σ(t = 1/2) only feels P or CP . As will be shown in detail there are important differences between the two symmetries CT and T , they will effectively gauge field configurations corresponding to untwisted/twisted sectors of closed strings and Neumann/Dirichlet boundary conditions of open strings. Not forgetting that our final aim is to orbifold/quotient our theory by gauging the discrete symmetries, let us proceed to check compatibility with the desired symmetries in detail. It is important to stress that field configurations satisfying any only feels P T or P which are isomorphic to Z 2 P T /P CT combinations of the previous symmetries exist, in principle, from the start in the theory. We can either impose by hand that the physical fields obey one of them (as is usual in QFT) or we can assume that we have a wide theory with all of these field configurations and obtain (self consistent) subtheories by building suitable projection operators | 3,068,636 | 119075145 | 0 | 16 |
that select some type of configurations. It is precisely this last construction that we have in mind when building several different theories out of one. In other words we are going to build different new theories by gauging discrete symmetries of the type P CT and P T . It is important to stress what the orbifold means in terms of the boundaries and bulk from the point of view of TM(GT). It is splitting the manifold M into two pieces creating one new boundary at t = 1/2. This boundary is going to feel only CP or P symmetries since it is located at the temporal fixed point of the orbifold. Figure 10 shows this procedure. In this way this new boundary is going to constrain the new theory in such a way that the boundary theories will correspond to open and unoriented versions of the original full theory. Tree Level Amplitudes for Open and Closed Unoriented Strings We start by considering tree level approximation to string amplitudes, i.e. the Riemann surfaces are of genus 0. These surfaces are the sphere (closed oriented strings) and its orbifolds: the disk (open oriented) and the projective plane (closed unoriented) as was discussed in section 2. From the point of view of TM(GT), orbifolding means that we split the manifold M into two pieces that are identified. As a result at t = 1/2, the fixed point of the orbifold, a new boundary is created. For different orbifolds we shall have different admissible field configurations. In the following | 3,068,637 | 119075145 | 0 | 16 |
discussion we studied which are the configurations compatible with P T and P CT for the several parity operations already introduced. Disk Let us start from the simplest case -the disk is obtained by the involution of the sphere under P 1 as given by (2.4). So consider the identifications under P 1 CT and P 1 T . For the first one the fields relate as The orientations of Σ andΣ are opposite. Under these relations the Wilson lines have the property This means that for the configurations obeying the relations (4.19) we loose the notion of time direction. Under the involution of our 3D manifold, using the above relations as geometrical identifications, the boundary becomes t = 0 and t = 1/2. For the moment let us check the compatibility of the observables with the proposed orbifold constructions given by the previous relations. In a very naive and straightforward way, when we use P 1 CT as given by (4.19) the charges should maintain their sign (q(t) ∼ = q(1 − t)). Then by exchanging boundaries we need to truncate the spectrum and set q ∼ =q = m in order the identification to make sense. Let us check what happens at the singular point of our orbifolded theory, t = 1/2. The fields are identified according to the previous rules but the manifold Σ(t = 1/2) = S 2 is only affected by P 1 . Take two Wilson lines that pierce the manifold in two distinct points, z and z ′ . | 3,068,638 | 119075145 | 0 | 16 |
Under the previous involution P 1 CT , z is identified withz for t = 1/2. Then, geometrically, we must have z ′ =z in order to have spatial identification of the piercings. The problem is that when we have only two Wilson lines, TM(GT) demands that they carry opposite charges. In order to implement the desired identification we are left with q = 0 as the only possibility. For the case where the Wilson lines pierce the manifold in the real axis, z = x and z ′ = x ′ , the involution is possible as pictured in figure 11 since we identify x ∼ = x and x ′ ∼ = x ′ . In the presence of three Wilson lines, following the same line of arguing, we will necessarily have one insertion in the boundary and two in the bulk as pictured in figure 12. Only in the presence of four Wilson lines, as pictured in figure 13 can we avoid any insertion in the boundary. Note that the identification B(z,z) ∼ = −B(z, z) in the real axis implies necessarily B(x, x) = 0. Remember that 2πn = B (see [32,39] for details). We could as well have an insertion in the boundary and one in the bulk This fact is simply the statement that by imposing P 1 CT we are actually imposing Neumann boundary conditions. The charges of the theory become q = m, this means that the string spectrum has only Kaluza-Klein momenta. Furthermore the monopole induced processes | 3,068,639 | 119075145 | 0 | 16 |
are suppressed, recall that they change the charge by an amount kn/2 which would take the charges out of the spectrum allowed in this configurations. Following our journey consider next P 1 T . The fields now are related in the following way P 1 T : The Wilson line has the same property (4.20) as in the previous case. Now the charges change sign under a P 1 T symmetry. As before identifying the charges in opposite boundaries truncates the spectrum, q(t) ∼ = −q(1 −t). So we must have q ∼ = −q = nk/4. We can, in this case identify two piercings in the bulk since the charge identifications are now q ∼ = −q is compatible with TM(GT). But we cannot insert any operator other than the identity φ 0 in the real axis since the corresponding charge must be zero q(x) = −q(x) = 0. Therefore this kind of orbifolding is only possible when we have a even number of Wilson lines propagating in the bulk. The result for two Wilson lines is pictured in figure 14 and for four in figure 15. Projective Plane We now consider the parity operation as the antipodal identification given in (2.8). We thus obtain the projective plane as the new 2d boundary of TM(GT). The transformation is given by discrete symmetry, that is t ′ ∼ = 1 − t, z ′ ∼ = −1/z andz ′ ∼ = −1/z. We obtain for P 2 CT Note that the relation between the integrals follows | 3,068,640 | 119075145 | 0 | 16 |
from taking into account the second and third equalities of (4.23), and the relations dz = dz ′ /z ′2 , dz = dz ′ /z ′2 , and consequently dz ∧ dz = −(1/z ′z′ )dz ′ ∧ dz ′ . Σ andΣ again have opposite orientations and are mapped into each other by the referred involution. Under these relations and in a similar way to (4.24) the action transforms under P 2 as given in (4.16) and any of the combinations P 2 CT or P 2 T keep it invariant. Also the Wilson lines have the same property given by (4.20). In the derivation of the previous identifications (4.23) we had to demand analyticity of the fields on the full sphere. This translates into demanding the transformation between the two charts covering the sphere to be well defined. Since ∂ u Λ = −z 2 ∂ z Λ and ∂ūΛ = −z 2 ∂zΛ the fields must behave at infinity and zero like If naively we didn't care about these last limits the relations would be plagued with Dirac deltas coming from the identity 2πδ 2 (z,z) = ∂ z (1/z) = ∂z(1/z). Once the previous behaviors are taken into account all these terms will vanish upon integration. Another way to interpret these results is to note that the points at infinity are not part of the chart (not physically meaningful), to check the physical behavior at those points we have to compute it at zero in the other chart. This time the | 3,068,641 | 119075145 | 0 | 16 |
charges compatible with P 2 CT are q = m since q ∼ =q. Once there are no boundaries it is not possible to have configurations with two Wilson which allow this kind of orbifold. In this way the lowest number of lines is four as pictured in figure 16. Furthermore the number of Wilson lines must be even. This configuration corresponds to untwisted closed unoriented string theories. Note that Λ, which is identified with string theory target space, is not orbifolded by P 2 CT . The charges allowed are q = m, the KK momenta of string theory. Once again the monopole processes are suppressed. For P 2 T the fields relate as In this case q = kn/4 since q ∼ = −q and further configurations with two Wilson lines are compatible with the orbifold as pictured in figure 17. In this case we have twisted unoriented closed strings. Note that the orbifold identifies Λ ∼ = −Λ such that the target space of string theory is orbifolded. The full construction, including the world-sheet parity, from the point of view of string theory is called an orientifold. The allowed charges q = kn/4 correspond to the winding number of string theory. The monopole processes are again crucial since allow, in the new boundary, the gluing of Wilson lines carrying opposite charges. We will return to this discussion. One Loop Amplitudes for Open and Closed Unoriented Strings Annulus We start with the already studied parity transformation Ω, as given by (2.15). There is nothing | 3,068,642 | 119075145 | 0 | 16 |
new to add to the fields relations (4.19) for P CT and (4.21) for P T , this time under the identifications t ′ = 1 − t, z ′ = −z andz ′ = −z. The resulting geometry is the annulus C 2 and has now two boundaries. For ΩCT the allowed charges are q = m due to the identification q ∼ =q and B(x) = 0 at the boundaries. We can have two insertions in the boundaries of the 2d CFT but not in the bulk due to the identifications of charges, basically the argument is the same as used for the disk. As in the disk we cannot have one single bulk insertion due to the total charge being necessarily zero in the full plane. Up to configurations with four Wilson lines we can have: two insertions in the boundary; one insertion in the bulk and one in the boundary corresponding to three Wilson lines; three insertions in the boundaries (with q = 0); one insertion in the bulk and two in the boundary corresponding to four Wilson lines; and two insertions in the bulk corresponding to four Wilson lines as pictured in figure 18. This construction corresponds to open oriented strings with Neumann boundary conditions. The charge spectrum is q = m, corresponding to KK momenta in string theory and the monopole induced processes are suppressed. It is Neumann because the gauged symmetry is of P CT type. We note that the definition of parity is not important, even for genus | 3,068,643 | 119075145 | 0 | 16 |
1 surfaces the results hold similarly to the previous cases for P 1 and P 2 used in genus 0. What is important is the inclusion of the discrete symmetry C! For ΩT the allowed charges are q = kn/4 due to the identification q ∼ = −q. There are no insertions in the boundary. One insertion in the bulk corresponds to two Wilson lines and two to four Wilson lines presented in picture 19. Möbius Strip Let us proceed to the parityΩ as given by (2.17). The results are pictured in figure 20 and are fairly similar. Note that it corresponds to two involutions of the torus with τ = 2iτ , one given by Ω resulting in the annulus, andã which maps the annulus into the Möbius strip. Then, for each insertion in the strip it is necessary to exist four in the torus. Once more we have forΩCT that B = −B = 0 in the boundaries and q is identified withq demanding the charges to be q = m, which correspond to the KK momenta of string theory. Due to this fact the monopole processes are suppressed in the configurations allowing this kind of orbifolding. This corresponds to Neumann boundary conditions. For theΩT case we have the identification of q with −q demanding the charges to be q = kn/4, the winding number of string theory. This time not allowing the monopole processes to play an important role, the charges are purelly magnetic. This corresponds to Dirichlet boundary conditions. As discussed in | 3,068,644 | 119075145 | 0 | 16 |
subsection 2.2 we can also consider the involution of the torus, with moduli τ = 1/2 + iτ 2 under ΩT or ΩCT . In this case four insertions in the torus correspond to two insertions in the strip as presented in figure 21 for the ΩT case. As previously explained both constructions result in the same region of the com-plex plane. Note that the resulting area in both cases is 2π 2 τ 2 and that in both cases the region [0, π] × i[0, 2πτ 2 ] is identified with the region [π, 2π] × i[0, 2πτ 2 ]. Again, for Ω ′ CT , we obtain q = m because q ∼ =q. The minimum number of insertions is two corresponding to four Wilson lines in the bulk. This construction corresponds to untwisted unoriented closed strings with only KK momenta in the spectrum. The monopole processes are suppressed. For Ω ′ T case we have q = kn/4 due to q ∼ = −q. We can have one single insertion in the bulk corresponding to two Wilson lines or two corresponding to four Wilson lines. This construction corresponds to twisted unoriented closed strings with only winding number. The monopole processes are present and are crucial in the construction. Note on Modular Invariance and the Relative Modular Group Modular invariance is a fundamental ingredient in string theory which makes closed string theories UV finite. What about the orbifolded theories? It is much more tricky. So if we actually want to ensure modular invariance we | 3,068,645 | 119075145 | 0 | 16 |
need to build a projection operator which ensures it. A good choice would be O = 2 + P T + P CT 4 (4.27) such that the exchange of orbifolds doesn't change it. This fact is well known in string theory (see [24] for details). In the case when we are dealing with orbifolds which result in open surfaces the modular transformation τ → −1/τ , according to the previous discussion, exchanges the boundary conditions (Neumman/Dirichlet). Note that orbifolding the target space in string theory (or equivalently the gauge group in TMGT) is effectively creating an orientifold plane where the boundary conditions must be Dirichlet (as for a D-brane). This is the equivalent of twisting for open strings. In terms of the bulk the modular transformation is exchanging the projections P CT ↔ P T . Let us put it in more exact terms. Consider some discrete group H of symmetries of the target space (or equivalently the gauge group of TMGT). Consider now the So far we have concentrated on one loop amplitudes only, i.e. genus 1 world-sheet surfaces orbifolds. For the pure bosonic case this is sufficient, but once we introduce fermions and supersymmetry new constraints emerge at two loop amplitudes. Specifically the modular group of closed Riemann surfaces at genus g is SL(2g, Z), upon orbifolding there is a residual conformal group, the so called Relative Modular Group [18] (see also [19][20][21]). For genus 1 this group is trivial but for higher genus it basically mixes neighboring tori, this means it mixes | 3,068,646 | 119075145 | 0 | 16 |
holes and crosscaps (note that any surface of higher genus can be obtained from sewing genus 1 surfaces). Furthermore, the string amplitudes defined on these genus 2 open/unoriented surfaces must factorise into products of genus 1 amplitudes. For instance a 2 torus amplitude can be thought as two 1 torus amplitudes connected trough an open string. For a discussion of the same kind of constraints for closed string amplitudes see [12][13][14][15][16]. The factorization and modular invariance of open/unoriented superstring theories amplitudes will induce generalized GSO projections ensuring the consistency of the resulting string theories. The correct Neveu-Schwarz (NS -antiperiodic conditions, target spacetime fermions) and Ramond (R -periodic conditions, target spacetime bosons) sectors were built from TMGT in [36]. There the minimal model given by the coset M k = SU(2) k+2 × SO(2) 2 /U(1) k+2 with the CS action (4.29) was considered. It induces, on the boundary, an N = 2 Super Conformal Field Theory (see also [33] for N = 1 SCFT). The boundary states of the 3D theory corresponding to the NS and R sectors are obtained as quantum superpositions of the 4 possible ground states (wave functions corresponding to the first Landau levelthe ground state is degenerate) of the gauge field B, that is to say we need to choose the correct basis of states. The GSO projections emerge in this way as some particular superposition of those 4 states at each boundary (for further details see [36]). It still remains to see how these constraints emerge from genus 2 amplitudes from | 3,068,647 | 119075145 | 0 | 16 |
TM and its orbifolds. We will discuss in detail these topics in some other occasion. Neumann and Dirichlet World-Sheet Boundary Conditions, Monopoles Processes and Charge Conjugation It is clear by now that the operation of charge conjugation C is selecting important properties of the new gauged theory. And here we are referring to the properties of the 2D boundary string theory. Gauging P CT results in having an open CFT with Neumann boundary conditions while, gauging P T results in having Dirichlet boundary conditions. So C effectively selects the kind of boundary conditions! In the case that P CT gives a closed unoriented manifold, we obtain an untwisted theory, while P T gives a twisted theory (orientifold X ∼ = −X). Again C effectively selects the theory to be twisted or not. These results are summarized in table 2. Twisted q = kn/4 q = kn/4 q = kn/4 q = kn/4 q = kn/4 Although these facts are closely related with strings T-duality, the C operation does not give us the dual spectrum. Upon gauging the full theory it is only selecting the Kaluza-Klein momenta or winding number as the spectrum of the configurations being gauged. From the point of view of the bulk theory the gauged configurations corresponding to Neumann boundary conditions correspond to two Wilson lines with one end attached to the 1D boundary of the new 2D boundary of the membrane at t = 1/2 and the other end attach to the 2D boundary at t = 0. For Dirichlet boundary conditions | 3,068,648 | 119075145 | 0 | 16 |
there is one single Wilson line with both ends in the 2D membrane boundary at t = 0 and a monopole insertion in the bulk of the 2D boundary at t = 1/2. Note that the Wilson lines do not, any longer, have a well defined direction in time, we have gauged time inversion. These results are presented in figure 25. For the case where we get unoriented manifolds the picture is quite similar. There are always an even number of bulk insertions. In the case of P CT the Wilson lines which are identified have the same charge, therefore there are no monopole processes involved. The two Wilson lines are glued at t = 1/2 becoming in the orbifolded theory one single line which has both ends attached to Σ 0 and one point in the middle belonging to Σ 1/2 . In the boundary CFT we see two vertex insertions with opposite momenta. This construction corresponds to untwisted string theories since the target space coordinates (corresponding to the gauge parameter Λ in TM(GT)) are not orbifolded. In the case of P T the identification is done between charges of opposite signs. Then two Wilson lines become one single line with its ends attached to Σ 0 , but at one end they have a q charge and in the other end they have a −q charge. In Σ 1/2 there is a monopole insertion which exchanges the sign of the charge. This construction corresponds to twisted string theories since the target space coordinates are | 3,068,649 | 119075145 | 0 | 16 |
orbifolded (Λ ∼ = −Λ). As a final consistency check in P CT the charges are always restricted to be q = m due to compatibility with the orbifold construction. By restricting the spectrum to these form we are actually eliminating the monopole processes for this particular configurations! T-Duality and Several U(1)'s The well know Target space or T-duality(for a review see [53]) of string theory is a combined symmetry of the background and the spectrum of momenta and winding modes. It interchanges winding modes with Kaluza-Klein modes. From the point of view of the orbifolded TM(GT) corresponding to open and unoriented string theories the projections P T truncate the charges spectrum to q = kn/4 (due to demanding q = −q) which in string theory is the winding number. The projections P CT truncate the charge spectrum to q = m (due to demanding q =q) which corresponds in string theory to the KK momenta. Note that P CT excludes all the monopole induced processes while P T singles out only monopole induced processes [32,37,39]. T-duality is, from the point of view of the 3D theory, effectively exchanging the two kinds of projections T − duality : P T ↔ P CT q = −q ↔ q =q (4.30) This is precisely what it must do. The nature of duality in 3D terms was discussed in some detail in [35]. It was shown there that it exchange topologically non trivial matter field configurations with topologically non trivial gauge field configurations. Although charge conjugation was not | 3,068,650 | 119075145 | 0 | 16 |
discussed there (only parity and time inversion), this mechanism can be thought as a charge conjugation operation. Note that C 2 = 1. It is also rather interesting that from the point of view of the membrane both T-duality and modular transformations are playing the same role. In some sense both phenomena are linked by the 3D bulk theory. So far we have considered only a single compact U(1) gauge group. But new phenomena emerge in the more general case. The extra gauge sectors are necessary any how [39]. Take then the general action with gauge group U(1) d × U(1) D with d U(1)'s noncompact and the remaining D's compact. Due to the charges not being quantize and the non existence of monopole-induced processes in the non compact gauge sector, the mechanism is slightly different (see section 3). But this operator can act as well over the noncompact sector. For the case of open manifolds M/P T , I ′ run over the indices for which we want to impose Neumann boundary conditions (on Λ I ′ ) and I ′′ over the indices corresponding to Dirichlet boundary conditions. For the case of closed manifolds M/P T the picture is similar but I ′ runs over the indices we want Λ I ′ to be orbifolded (obtaining an orientifold or twisted sector). In the case of several U(1)'s more general symmetries (therefore orbifold groups) can be considered (for instance Z N ). Those symmetries are encoded in the Chern-Simons coefficient K IJ . Conclusion and | 3,068,651 | 119075145 | 0 | 16 |
Discussion In this paper we have shown how one can get open and closed unoriented string theories from the Topological Membrane. There were two major ingredients: one is the Horava idea about orbifolding, the second is that the orbifold symmetry was a discrete symmetry of TMGT. The orbifold works from the point of view of the membrane as a projection of field configurations obeying either P T or P CT symmetries (the only two kinds of discrete symmetries compatible with TMGT). For P CT type projections we obtained Neumann boundary conditions for open strings and untwisted sectors for closed unoriented strings. For P T type projections we obtained Dirichlet boundary conditions for open strings and twisted sectors for closed unoriented strings. For P CT q =q = m, so only the string Kaluza Klein modes survive. In this case the monopole induced processes are completely suppressed. For P T q = −q = kn/4, so only the string winding modes survive. In this case only monopole induced processes are present, being the charges purelly magnetic. Charge conjugation C plays an important role in all the processes playing the role of a Z 2 symmetry of the string theory target space. These results can be generalized to symmetries of the target space encoded in the tensor K IJ and are closely connected, both with modular transformations and T-duality which exchange P T ↔ P CT . This work is the first part of our study of open and unoriented string theories. In the second part [54] we | 3,068,652 | 119075145 | 0 | 16 |
shall derive the partition functions of the boundary CFT from the bulk TMGT [55][56][57][58][59]. Also an important issue to address in future work will be to generalize the constructions presented here to non trivial boundary CFT's [54], for example WZNW models and different coset models which can be obtained from TM with non-Abelian TMGT. As a final remark let us note that the string photon Wilson line has been left out. TM(GT) can take account of it as well: for any closed Σ there is a symmetry of the gauge group coupling tensor K IJ → K IJ + δ I χ J − δ J χ I where each χ I = χ I [A] is taken to be some function of the A I 's. This transformation affects only B IJ and the induced terms vanish upon integration by parts. Once we consider the orbifold of the theory the new orbifolded Σ o has a boundary and the induced terms will not vanish any longer but induce a new action on the boundary ∂Σ o , they will be precisely the new gauge photon action of open string theories. As is well known the choice of the gauge group of string theory, i.e. the Chan-Paton factors structure carried by this photon Wilson line will be determined by the cancellation of the open string theory gauge anomalies (see [24] and references therein). We postpone the proper treatment of this issue from the point of view of TM to another occasion [54]. supported by PRAXIS XXI/BD/11461/97 | 3,068,653 | 119075145 | 0 | 16 |
grant from FCT (Portugal). The work of IK is supported by PPARC Grant PPA/G/0/1998/00567 and EUROGRID EU HPRN-CT-1999-00161. | 3,068,654 | 119075145 | 0 | 16 |
Biophysical basis underlying dynamic Lck activation visualized by ZapLck FRET biosensor A new biosensor was engineered to visualize Lck kinase preactivation and regulation in live T cells. INTRODUCTION T cell receptor (TCR) signals are initiated when foreign antigens are presented to induce the TCR complex formation (1,2). The first detectable event of TCR signals is that an activated Src family kinase (SFK), lymphocyte-specific protein tyrosine kinase (Lck), phosphorylates the tyrosine residues in the immunoreceptor tyrosine-based activation motifs (ITAMs) located at the cytoplasmic tails of TCR z chain subunits and co-receptor CD3 (1,2). Phosphorylated ITAMs function as docking sites to recruit the cytoplasmic kinase Zap70 (z chain-associated protein 70 kDa) to the plasma membrane for phosphorylation and activation by Lck (3). Active Zap70 then phosphorylates other kinases to transduce downstream signaling cascades, such as the activation of extracellular signal-regulated kinase (ERK) to up-regulate interleukin-2 (IL-2) expressions (4)(5)(6). In addition to CD3 stimulation, CD28 costimulation can promote the enrichment of membrane molecular clusters and facilitate T cell proliferation and differentiation (7,8). As an early kinase activated upon TCR triggering, Lck plays crucial roles in T cell activation and development (9,10). Abnormal Lck expression, activation, and transport have been reported to cause severe immune deficiency in humans (11,12). Overexpression of Lck has been identified in cancers such as lymphoma and leukemia (13). However, the pathophysiological regulations of Lck function at the initiation of TCR signaling remain largely unclear. As an SFK, Lck has a similar protein structure as other SFK members, containing an N-terminal-specific domain, Src homology 3 (SH3), | 3,068,655 | 195186884 | 0 | 16 |
SH2, a kinase domain, and a C-terminal tail (14,15). These domains regulate Lck functions via its molecular conformation, kinase-domain accessibility, TCR interaction, and plasma membrane localization (16,17). Particularly, residues lysine 287 (K287) and tyrosine 394 (Y394) within the Lck kinase domain as well as tyrosine 505 (Y505) at the cy-toplasmic tail are crucial regulators of Lck kinase activity (18,19). While modifications of K287 and Y505 have been shown to have activating and inhibitory effects on Lck activity, respectively, the role of Y394 remains unclear. Phosphorylation of Y394 has been suggested to stabilize the open active conformation of Lck and positively contribute to Lck kinase activation, and the tyrosine-to-phenylalanine mutation Y394F has been shown to impair Lck activity (18,20,21). However, it has remained controversial how the mutation on Y394 affects Lck kinase activity and whether LckY394F (LckYF) is enzymatically incapable of being activated (21)(22)(23)(24). It is also unclear how Y394 phosphorylation affects the biophysical properties that mediate Lck activation and TCR downstream signals (25,26). Fluorescence (or Förster) resonance energy transfer (FRET)based biosensors have been widely used to visualize molecular activities in live cells with high spatiotemporal resolution (27,28). However, there are currently no broadly applicable Lck biosensors to allow the convenient visualization of the dynamic kinase activity of Lck in live T cells. A fluorescence lifetime imaging microscopy (FLIM)-based, full-length Lck, FRET biosensor has been developed and applied to investigate the regulatory effect of conformational change in Lck, although the biosensor is not able to directly report Lck kinase activity (26,29). In the current study, we developed | 3,068,656 | 195186884 | 0 | 16 |
a new substratebased single-chain Lck FRET biosensor (ZapLck) to monitor the spatiotemporal Lck kinase activity in live T cells. This Lck biosensor contains an enhanced cyan fluorescence protein (ECFP) and a variant of yellow fluorescence protein (YPet) as the reporting unit and an SH2 domain and Lck-sensitive tyrosine peptide as the sensing unit (30)(31)(32). Being substrate based without an enzymatic domain, our biosensor should not substantially perturb endogenous Lck signals in T cells. With ZapLck, we found that the putatively inactive mutant LckY394F diffused faster than wild-type Lck (LckWT) and exhibited low kinase activity at the basal level. However, both LckY394F and LckWT can be fully activated upon the ligation and clustering of TCRs in live T cells. Further experiments revealed that LckY394F can be activated to the same degree as LckWT and can function to transduce the TCR signal downstream to ERK activation with similar strength and kinetics as LckWT. As such, ZapLck provided a powerful tool to unravel the molecular basis of Lck activation in live cells. Engineering and in vitro characterization of a new Lck-FRET-Zap70FY kinase biosensor On the basis of the principle of FRET, we used a previously designed kinase biosensor cassette for the construction of the Lck FRET biosensors (31,32). This cassette consists of an ECFP as the donor, an Src SH2 domain with a C185A mutation, a flexible linker, an interchangeable tyrosine substrate of Lck kinase, and a YPet as the acceptor, from the N to the C terminus ( Fig. 1A) (31). In this cassette, a biosensor in the | 3,068,657 | 195186884 | 0 | 16 |
resting state is expected to show a high FRET efficiency between the donor and acceptor due to their sticky interface and, hence, a low ECFP/FRET emission ratio (Fig. 1A). We used a mutant Src SH2 domain that has been optimized to bind phosphor-tyrosine-containing peptides with high affinity and reversibility (33). Upon Lck kinase activation, the substrate is phosphorylated and binds to the SH2 domain, causing ECFP and YPet to separate and resulting in a high ECFP/FRET emission ratio (Fig. 1A). Therefore, we will use the ECFP/FRET ratio reported by the biosensor to represent the Lck kinase activity. To design a highly sensitive and specific Lck biosensor, we identified the optimal tyrosine substrate among known substrates of Lck in sensing its kinase activity. Upon TCR triggering in T cells, active Lck can phosphorylate the D/ExxYxxI/Lx (6)(7)(8)(9)(10)(11)(12)YxxI/L peptide sequences on TCR z chain ITAMs (34). The double-phosphorylated ITAM tyrosine peptide then acts as the docking site to recruit an Syk family kinase, Zap70, to the plasma membrane, where Lck then phosphorylates Zap70 residues tyrosine 315 (Y315) and tyrosine 319 (Y319), inducing Zap70 activation and further phosphorylation (35,36). The Zap70 residue Y319 has also been reported to be essential for TCR signaling, while Y315 is nonessential (37). Hence, we engineered Lck biosensors with substrate peptides containing the Y315 and Y319 tyrosine sites (Zap70YY), or a single mutation of Y315F to promote the primary SH2 binding site pY319 and simplify the design (Zap70FY; Fig. 1B). Among three ITAMs located on the TCR z chains, the C-terminal motif is sensitive to | 3,068,658 | 195186884 | 0 | 16 |
Lck phosphorylation and most likely recruits Zap70 upon phosphorylation (38). Since there are two tyrosine residues on the C-terminal motif, two separate substrates encompassing either the first or second tyrosine site were designed (ITAM1 and ITAM2) (Fig. 1B). As Lck can autophosphorylate the Y394 site, we designed another substrate peptide containing Y394 from Lck ( Fig. 1B) (14,24). The S2D). Together, we conclude that the Lck-FRET-Zap70FY biosensor is a good candidate biosensor to monitor Lck kinase activation in live cells with reasonable sensitivity and specificity. Lck plays crucial roles in TCR signaling transduction in T cells (39,40), so we further characterized our Lck-FRET-Zap70FY biosensor in T cells by comparing it to a negative control biosensor with dual mutations Y315F and Y319F (Lck-FRET-Zap70FF). We stimulated JurkatE6-1 (Jurkat) cells with conjugated clusters of CD3/CD28 antibodies to initiate TCR signaling and activate Lck (7,8). Upon stimulation, the ECFP/FRET ratio of the Lck-FRET-Zap70FY biosensor increased to the peak value within 6 min and then leveled off. This activation can be inhibited by PP1, indicating that the ratio change was specific to SFK kinase activity (Fig. 2, A and B, fig. S3, and movie S1). In contrast, the ECFP/FRET ratio of the mutant biosensor, Lck-FRET-Zap70FF, did not change after stimulation, confirming that the biosensor ratio change is mediated by tyrosine phosphorylation as designed (Figs. 1A and 2, A and B) (37). As quantified in Fig. 2C, antibody stimulation induced a significant~15% ratio increase of our Lck-FRET-Zap70FY FRET biosensor in Jurkat cells, which can be inhibited by PP1 to~28% less than the | 3,068,659 | 195186884 | 0 | 16 |
basal level. As such, our Lck-FRET-Zap70FY biosensor is sensitive and specific for monitoring Lck kinase activity in live T cells. Hereafter, we refer to the Lck-FRET-Zap70FY biosensor as ZapLck. . Error bars: mean ± SEM. Two-tailed Student's t test was used for statistical analysis. ***Significant difference from other groups in the same cluster or from the indicated group in the other cluster, P < 1 × 10 −3 . A representative Jurkat cell under treatment is also shown in movie S1. Assuming that Lck was fully activated by stimulation until it was inhibited by PP1, we use the following formula to estimate the portion of preactivated Lck kinase (Fig. 2C) Ratio Basal À Ratio Inhibitor Ratio Peak À Ratio Inhibitor  100% It was therefore estimated that 62% of Lck was preactivated in Jurkat cells at the basal level before stimulation, although it is possible that PP1 inhibition was not sufficient to reduce Lck activity to the lowest level after Jurkat activation. We performed PP1 inhibition and a washout experiment to confirm the basal Lck FRET ratio in Jurkat cells ( fig. S4). Jurkat cells directly treated with PP1 showed a ratio value of 0.249 ± 0.007 ( fig. S4), similar to the that of the CD3/ CD28-stimulated Jurkat cells treated by PP1 (0.246 ± 0.004; Fig. 2C), which was used to estimate the preactivated fraction of Lck. This control experiment confirmed that the activation/inhibition assay can be used to estimate the preactivated portion of the Lck kinase with reasonable accuracy, except for the potential limitation | 3,068,660 | 195186884 | 0 | 16 |
that the relationship between FRET ratio signal and amount of active Lck molecule may not be direct. Further experiments show that PP1 inhibition can also significantly reduce the Lck FRET signal in primary peripheral blood mononuclear cells (PBMCs) ( fig. S4), indicating that the preactivated portion of the Lck kinase also exists in primary T cells. Furthermore, the different preactivated portions between the cells expressing LckWT-mCherry and LckY394F-mCherry are probably not due to their different subcellular distributions, since the mCherry-tagged LckWT showed similar subcellular localization as mCherry-LckYF ( fig. S5), as well as the endogenous Lck revealed by immunostaining (41). Both the LckWT and LckY394F kinases can be activated to a similar level in live T cells by TCR co-receptor stimulation To investigate the regulation of Lck kinase function in T cells, we used the new ZapLck biosensor to image Lck-deficient J.Cam1.6 (JCam) T cells derived from Jurkat. When the JCam cells were stimulated with CD3/CD28 antibody clusters, no ECFP/FRET emission ratio change of the biosensor was detected, confirming that this biosensor can be applied to specifically monitor Lck kinase activity in T cells (Fig. 3, A to C, and movie S1). We then infected the JCam cells with lentiviral LckWT, the kinasedead Lck with K273R mutation (LckKR), or LckY394F and named the resulting cells JCam-LckWT, JCam-LckKR, and JCam-LckYF, respectively (20,21,42). When these cells coexpressing ZapLck biosensors were stimulated with CD3/CD28 antibody clusters, the JCam-LckWT cells showed an average 19% increase in ECFP/FRET emission ratio from the basal level, while JCam-LckYF cells showed an unexpected 40% | 3,068,661 | 195186884 | 0 | 16 |
increase, which reached to a peak value similar to the biosensors in JCam-LckWT cells (Fig. 3). As expected, JCam-LckKR showed no ratio change (Fig. 3, A and B). It is noteworthy that JCam cells expressed LckWT at a low level with an mCherry intensity ranging between 500 and 1500, while LckYF was expressed at a high level with an intensity range of 500 to 8000. To rule out the possibility that the expression levels of the Lck variants may affect the results in Fig. 3, we further quantified time courses of the Lck ECFP/FRET ratio in JCam cells expressing different mCherry-tagged Lck mutants at comparable levels (500 to 1500 in intensity; fig. S3). At this expression level, JCam cells with Lck mutants show similarly different dynamic responses as before ( Fig. 3 and fig. S3E). Therefore, these results confirmed that the observed differential biosensor responses in Fig. 3 were mainly caused by different types of Lck variants but not by their expression levels ( fig. S3E). Together, our results indicate that, similar to LckWT, LckY394F has a kinase activity that can be significantly activated by CD3/CD28 (LckWT), 39.9 ± 1.6% (LckY394F), 0.4 ± 0.3% (LckKR), and 3.7 ± 1.4% (JCam). The Bonferroni multiple comparison test provided by the multcompare function in the MATLAB statistics toolbox was used for statistical tests. ***Significant difference from all other groups with P < 1 × 10 -3 . Representative JCam cells with or without different Lck mutants under treatment are also shown in movie S1. co-receptor stimulation in JCam cells. | 3,068,662 | 195186884 | 0 | 16 |
PP1 treatment significantly inhibited the observed increase of the ECFP/FRET ratio in JCam-LckWT and JCam-LckYF cells (Fig. 3, A and B, and fig. S3), indicating that these signals are caused by Lck kinase activation. Although both c-Src and Fyn can activate the Lck biosensor in vitro (fig. S1), the expression of c-Src and Fyn is considered low in T cells (39). Less c-Src and Fyn than Lck were expressed in Jurkat T cells (43). Our Lck biosensor showed activation in stimulated JCam cells reconstituted with LckWT and LckYF, but not in the Lck-deficient JCam cell, although these JCam cells showed similar levels of c-Src/Fyn expression as that of Jurkat cells (Fig. 3) (43). These results suggest that the FRET response of our Lck biosensor is specific to Lck kinase in T cells. Our ZapLck biosensor, hence, provided a new tool in revealing the preactivated portions of Lck. A previously published Src biosensor showed an overall very low ECFP/FRET ratio and did not detect any preactivated kinase in Jurkat cells, although this Src sensor is functional in detecting Src kinase activations since it responded strongly to a pervanadate stimulation in HeLa cells in the control experiment ( fig. S5) (31). Similar to Jurkat cells, before stimulation, the basal Lck activity of JCam-LckWT was significantly higher than that of JCam-LckYF, JCam-LckKR, and JCam cells (Fig. 3, A, B, and D, and movie S1). When treated with PP1, the emission ratio of the biosensor in JCam-LckWT cells, but not that in JCam-LckYF, dropped to a lower level than the | 3,068,663 | 195186884 | 0 | 16 |
basal ratio. Again, assuming that Lck was fully activated by the antibodies and PP1 inhibited all Lck activity, we estimate that 51% of active LckWT were preactivated at the basal level in JCam cells, which is consistent with our observation in Jurkat cells and a previous report (44). In contrast, only~2% of LckY394F were preactivated, which probably explains why some of the previous publications found LckY394F inactive in T cells, although LckY394F had been reported with kinase activity in Escherichia coli (Fig. 3, B and E) (22,23). The Western blot results further confirmed that LckWT can be quite active with or without co-receptor stimulation, while LckY394F was activated only after stimulation ( fig. S3). To examine whether the Zap70 residue Y319 on the biosensor was phosphorylated by the endogenous Zap70 (37), we performed control experiments in the Zap70-deficient P116 cells ( fig. S3F) (45). The Zap70-deficient P116 cells showed similar Lck activation and biosensor responses to CD3/CD28 stimulation as the Jurkat and JCam-LckWT cells, which can be significantly reduced by PP1 inhibition (Figs. 2 and 3 and fig. S3F). These results confirmed that the biosensor responses are unlikely due to the unspecific activation by Zap70. Together, our ZapLck biosensor showed that LckWT, but not LckY394F, is preactivated in T cells. Although LckY394F has a lower basal activity than LckWT, it can be fully activated upon the ligation and clustering of TCR and co-receptors, suggesting that the Y394 site is crucial for the high Lck kinase activity only under the basal condition, but dispensable in activated T | 3,068,664 | 195186884 | 0 | 16 |
cells. LckWT and LckY394F transduce fast ERK activation downstream Since the mutant LckY394F can still respond to TCR co-receptor stimulation, we further examined its ability to transduce signals downstream. We used a FRET-based ERK (NES) biosensor residing outside the nucleus (46) to monitor the dynamic ERK activation in T cells with different Lck mutants, with the FRET/ECFP emission intensity ratio of the ERK biosensor representing ERK activity. JCam-LckWT/-LckYF/-LckKR and Jurkat cells expressing ERK biosensors were monitored before and after antibody stimulation. Before stimulation, JCam-LckWT and Jurkat cells had higher basal FRET/ECFP ratio than the other groups (Fig. 4, A and D, and movie S2). After stimulation, the quantified FRET/ECFP ratio in JCam-LckWT, JCam-LckYF, and Jurkat cells increased to a similar level within 10 min, which can then be partially inhibited by PP1 (Fig. 4, fig. S3B, and movie S2). These results indicate that LckY394F can transduce TCR signals downstream to ERK activity with similar efficiency as that of LckWT. Unexpectedly, the normalized FRET/ECFP ratio curves show that ERK can also be activated in the Lck-deficient JCam and JCam-LckKR cells (Fig. 4, C and F). Further analysis of the ERK activation kinetics showed that the ERK kinase was activated significantly slower in cells without functional Lck kinases than those with LckWT or LckY394F (Fig. 4, C and G), suggesting the role of the Lck kinase in mediating the fast ERK response upon activation. When the cells were treated with PP1, ERK activity decreased close to basal level in all groups, indicating that the observed slow ERK activation | 3,068,665 | 195186884 | 0 | 16 |
in Lck kinase-deficient cells may be mediated by an SFK member other than Lck (Fig. 4, B and C). In addition, when treated with PP1, the observed ERK kinase activity in JCam-LckWT dropped below the basal level. Accordingly, 61% of the ERK kinase was estimated to be preactivated in JCam-LckWT cells (Fig. 4, B and C). We further examined the level of endogenous phosphor-ERK in activated Jurkat and JCam cells with different Lck mutants ( fig. S6). Western blot results showed that within 5 min of CD3/CD28 activation, significantly more phosphor-ERK can be detected in Jurkat, JCam-LckWT, and JCam-LckYF cells, but not in the JCam-LckKR cells ( fig. S6). Therefore, we conclude that downstream of CD3/CD28 activation, the ERK kinase can be activated in both Lck kinase-dependent and Lck kinase-independent pathways, with Lck or LckY394F kinase mediating a fast ERK activation, while an SFK member other than Lck contributes to the slow phase of ERK activation. Lck activity is important for the rapid activation of downstream ERK signals, but the role of preactivated Lck seems unclear here (Fig. 4G). Overall, these results also verify that LckY394F upon stimulation has similar functionality as LckWT in mediating the activation of downstream T cell signaling. CD3 is primarily responsible for the LckYF activation facilitated by CD28 costimulation The high response level of JCam-LckYF cells allows us to explore the concentration-dependent kinetics of costimulation signals, as well as delineate roles played by CD3 and CD28 ligations. We, hence, stimulated the JCam-LckYF cells with different concentrations of combined CD3/ CD28, CD3-only, | 3,068,666 | 195186884 | 0 | 16 |
or CD28-only antibody clusters while monitoring the Lck kinase activation dynamics with the ZapLck biosensor. Since multiple antibody combinations and concentrations were involved in these experiments, for simplicity, we defined the standard concentration of antibodies used in Figs. 2 to 4 as "1 unit" (10 and 5 mg/ml for CD3 and CD28 antibodies, respectively; see Fig. 5). As a result, the lower concentration of "1/10 unit" corresponds to 1 mg/ml and 0.5 mg/ml for CD3 and CD28 antibodies, respectively, while "1/50 unit" represents 0.2 mg/ml and 0.1 mg/ml for CD3 and CD28 antibodies, respectively. All the antibodies were premixed and preclustered before treating cells as detailed in Materials and Methods. As shown in Fig. 5 (A and B) and fig. S7, the quantified time courses of Lck activation show that different concentrations of CD3/CD28 antibodies or CD3-only antibody can all activate the Lck kinase with an average response of about 30% from the basal level. The magnitude of Lck kinase activation (normalized peak ratio) was relatively independent of the concentrations of combined CD3/CD28 antibodies, while the CD3-only groups showed significantly lower magnitude and slower kinetics at 1/50 unit (Fig. 5, A to C). The CD28-only groups did not show detectable Lck activation even at a high level of antibody treatment at 1 unit ( fig. S8). Several measurements were calculated from the single-cell time courses to evaluate the activation kinetics and stability of Lck kinase signals ( fig. S7). The activation halftime was calculated as the time gap between stimulation to when FRET ratio reaches half | 3,068,667 | 195186884 | 0 | 16 |
of its maximum in single cells, which is used to evaluate the activation speed of cellular Lck activation (Fig. 5C). Our results show that antibody concentration has a significant effect on the activation speed of the Lck kinase. The activation halftime almost tripled when the concentration was reduced from 1 to 1/50 for both CD3/CD28 and CD3-only antibody cluster stimuli (Fig. 5C). Judging from the time courses, the longer activation halftime is caused by both delayed response and slow activation kinetics at low-concentration stimuli (Fig. 5 and fig. S6). Furthermore, removing CD28 antibodies from the stimuli increased the activation halftime only at the lowest concentration of 1/50 (Fig. 5C). Together, our results indicate that CD3 is essential, while CD28 is supplementary, for Lck activation in T cells. This effect of CD28 on activation magnitude and speed only becomes prominent at the lowest antibody concentration of 1/50 (Fig. 5). Our results also show that antibody concentration mainly regulates the activation speed, but not the magnitude, of Lck kinase signals. The kinase activities of LckWT and LckY394F are distinctively regulated at the basal level We then asked why LckY394F has a basal activity distinctive from that of LckWT but both can reach a similar level in activities upon stimulation. Previous studies reported that Lck is preactivated and forms clusters in live T cells (26,44). Our results also confirmed that LckWT, but not LckY394F, was preactivated in Jurkat and when expressed in JCam cells. Therefore, we hypothesize that the kinase activities of LckWT and LckY394F may be differentially regulated | 3,068,668 | 195186884 | 0 | 16 |
by their clustering or Lck-Lck interaction in T cells. Hypothesizing Lck expression levels and hence their surface densities may affect Lck-Lck interaction, we investigated the effect of the expression level of Lck on its basal activity. Considering that Lck localizes to plasma membrane through its N terminus, we fused mCherry to the C terminus of Lck to obtain an Lck-mCherry construct, such that the expression level of Lck is represented by the mCherry fluorescence intensity. JCam cells expressing LckY394F-mCherry or LckWT-mCherry were cotransfected with the ZapLck biosensor to monitor the basal Lck activities by the FRET signals. By quantifying the basal-level ECFP/FRET emission ratio versus mCherry intensity in the same single cells, we modeled the relation between the biosensor emission ratio (reflecting Lck kinase activity) and the mCherry intensity (reflecting the copy number of Lck or LckYF) in JCam cells expressing LckWT or LckY394F. JCam-LckWT-mCherry cells showed much lower expression level than the JCam-LckYF-mCherry cells, indicating that the expression level of LckWT, but not that of LckY394F, is strictly regulated in T cells (Fig. 6). We assume that the JCam cells without Lck expression (mCherry fluorescence intensity, FI = 0) have a minimal ECFP/FRET ratio of the value b1. Accordingly, we formulate a coupled model between the biosensor emission ratio and mCherry intensity for both JCam-LckWT and JCam-LckYF cells (Fig. 6C). This model predicts a ratio value of b1 at zero mCherry intensity for both cell groups. The results revealed a nonlinear relation between Ratio WT and Intensity WT in JCam-LckWT cells (the exponential power is | 3,068,669 | 195186884 | 0 | 16 |
the best fit at p2 = ½) but a linear relation between Ratio YF and Intensity YF in JCam-LckYF cells (p3 = 1, with balanced residuals) (Fig. 6, A to C). Therefore, we conclude that the correlation between LckWT kinase activity and expression is nonlinear, featured by a relatively faster increase of activity as the expression level increases in low-Lckexpressing cells but slower in high-expressing cells. In contrast, the activity of LckYF increases linearly as its expression level increases. The overall LckWT activity is also twice more sensitive to the expression level than that of LckYF (Fig. 6D). In these experiments, cells were selected into low-intensity (bottom 20 percentile) and high-intensity (top 20 percentile) groups, and ordered by intensity values in each group. A pairwise DRatio/DIntensity value was then calculated for two consecutive cells in the groups to evaluate the sensitivity of Lck kinase activity to expression level. These results suggest that LckY394F activity is less sensitive to expression level. Potentially, such an outcome could be explained by decreased interactions between LckY394F monomers compared with those of LckWT. In contrast, LckWT had a high basal kinase activity, which is significantly and nonlinearly affected by the expression level, suggesting a possible role of Lck-Lck interaction mediated by tyrosine phosphorylation. Together, these results indicate that the basal kinase activities of LckY394F and LckWT are distinctively regulated by their expression level, possibly via different modes of intermolecular interaction. We reasoned that the distinctive states of LckWT and LckY394F in interacting and forming complexes can lead to a difference in | 3,068,670 | 195186884 | 0 | 16 |
their diffusion rates. We, hence, evaluated the diffusion rate of LckWT and LckY394F in live T cells with the fluorescence recovery after photobleaching (FRAP) experiment and finite element model analysis established in our laboratory (47). JCam-LckWT-mCherry and JCam-LckYF-mCherry cells with or without CD3/CD28 stimulation were photobleached and monitored for mCherry recovery (Fig. 7A). The images during recovery were used as input for a finite element diffusion model, which allows the quantification of Lck diffusion rate in live cells, relatively independent of cell shape, photobleach pattern, and initial intracellular fluorescence distribution (47,48). The estimated apparent diffusion rates of LckWT and LckY394YF with and without activation lie in the range of 0.60 to 2.13 mm 2 /s, which is consistent with previous reported diffusion rates of Src and free lipid (Fig. 7) (49, 50). We found that at the basal level, LckWT diffused about two times slower than LckY394F (Fig. 7), indicating that LckWT may be significantly more aggregated than LckY394F. After stimulation, however, LckWT and LckY394F both had similar slow diffusion rates and diffused significantly slower than their basal levels (Fig. 7D). LckWT and LckY394F may form large-scale molecular complexes after stimulation and reduce their mobility at the plasma membrane of T cells. The diffusion speed of LckWT and LckY394F appears to be inversely related to their respective kinase activity, indicating that the level of Lck-Lck interaction may reflect and possibly regulate their kinase activities in live T cells (Figs. 3 and 8). Since diffusion speed is an indicator of the interacting state of a molecule at | 3,068,671 | 195186884 | 0 | 16 |
the plasma membrane, we performed additional imaging experiments with LckWT and LckY394F tagged with bimolecular fluorescence complementation (BiFC) split Venus (the N-terminal VN173 and C-terminal VC155 fragment of Venus, a yellow fluorescence protein) to compare the interacting states of Lck with or without the Y394F mutation, and before and after CD3/CD28 stimulation (Fig. 8). Specifically, Lck-Lck interaction can cause the binding between VN173 and VC155, which completes the formation of Venus to produce fluorescence signals after about 50 min of chromophore maturation (Fig. 8A). The JCam cells expressing LckWT-or LckYF-split Venus were stimulated with CD3/CD28 antibody clusters. Imaging and quantification results showed that both group of cells had significant increase in Venus intensity after stimulation, with a lower background and larger increase of dimerization detected in LckY394F cells than in the LckWT group (Fig. 8, B and C). These results showed that CD3/ CD28 activation can enhance dimerization of LckY394F and LckWT in JCam cells. Therefore, Lck-Lck interactions were positively correlated to kinase activities of LckWT and LckY394F in live T cells (Figs. 3 and 8). Together, we conclude that the expression and interaction of Lck are tightly regulated in T cells. LckWT has a high level of aggregation, which may contribute to its kinase activity before CD3/CD28 stimulation. In contrast, LckY394F molecules remain relatively unbound and inactive at basal levels. Both LckWT and LckY394F may form complexes and become activated with slow diffusion rates following CD3/CD28 stimulation. DISCUSSION Single-chain substrate-based FRET biosensors have been widely used to provide real-time visualization of posttranslational modification dynamics in | 3,068,672 | 195186884 | 0 | 16 |
live cells (27,28). With a fixed (1:1) donor-to-acceptor ratio, the signals from these biosensors can be conveniently quantified by taking the ratio between donor and acceptor intensity. Biosensors designed to monitor enzymes typically contain a targeting substrate, but not enzymatic domains, so they should not cause significant perturbation of endogenous signals when expressed in live cells. As a result, these biosensors can function as a "real-time immunostaining assay" in single live cells to provide accurate and dynamic readout on the phosphorylation status of a selected substrate peptide, rendering a powerful imaging tool complementary to immunostaining and biochemical assays, which either require the killing of cells or only measure the average signal of a population of cells. On the basis of the design principle for single-chain and substrate-based FRET sensors (Fig. 1A), our group has successfully engineered and characterized several biosensors to visualize the kinase activity of Src, FAK, and Zap70 (31,32,51), some of which use our highly sensitive FRET pair, ECFP and YPet, for enhanced dynamic range in signals. With a similar strategy, here we engineered the new Lck biosensor by rationally examining several known tyrosine substrate peptides of the Lck kinase, including those from Zap70, ITAM, and Lck itself. The optimized ZapLck biosensor contains a Zap70 tyrosine peptide with a single tyrosine-to-phenylalanine mutation, which is identified as the most sensitive and specific among all substrates, with about 40% increase of signals in activated T cells. Using this new biosensor, we investigated the molecular mechanism underlying the Lck activation in live T cells. Some reports suggest | 3,068,673 | 195186884 | 0 | 16 |
that there may be just~2% of Lck with phosphorylated Y394 in resting T cells; however, other groups argue that there may be up to~40% Lck phosphorylated on Y394 (44,52). Phosphorylation of Y394 has been widely used as an assay to probe Lck kinase activity since there is a lack of Lck kinase reporters so far. Through signals reported by the ZapLck FRET biosensors, we found that up to 62 and 51% of Lck kinase were preactivated in Jurkat and JCam-LckWT cells, respectively, while LckY394F almost completely abolished the preactivation in JCam cells. As such, our results support the notion that a substantial fraction of Lck should be preactivated with phosphorylated Y394. It is possible that Lck preactivation is facilitated by the interaction between free Lck, ITAMs, and Zap70 (53,54), with phosphorylated LckY394 serving as a docking residue. The mutation Y394F may affect the binding affinity and, hence, significantly reduce the basal-level Lck activation. The overall Lck kinase activity is regulated by the phosphorylation dynamic equilibrium between the Y505 and Y394 tyrosine residues (55). The Y394F mutation has been reported to either completely abolish or decrease the Lck kinase activity to a very low level (21,23,26). However, our ZapLck biosensor visualized an increase of LckY394F kinase activity to be~92% of LckWT activation upon CD3/CD28 ligation and clustering in live JCam cells (Fig. 3). Hence, our results suggest that LckY394F can be fully functional in its kinase activities upon CD3/CD28 clustering, possibly via enrichment and sequestration of CD3/CD28 and LckY394F at the same submembrane compartments mediated by actin | 3,068,674 | 195186884 | 0 | 16 |
polymerization and raft formation (56). This kinase activation of LckY394F has not been observed in T cells previously, probably due to the lack of precise dynamic assays and methods in single T cells (21,23,26). The ZapLck biosensor reported a 40% increase of Lck kinase activity in JCam-LckYF cells upon co-receptor ligation. This dynamic range allowed us to investigate concentration dependence and individual contribution of co-receptors, and delineate the different roles played by CD3 and CD28 receptors. As such, these findings highlight the exquisite sensitivity of our ZapLck biosensor in detecting Lck kinase activity in single live T cells. To further explore the effect of mutant LckY394F on downstream TCR signals, we used the ERK biosensor to monitor ERK kinase activity, which stimulates the expression of IL-2 and nuclear factor kB. We found that Lck kinase-dependent ERK activation responds faster to stimulation than ERK activation independent of Lck. Therefore, our data suggest that there are two signaling pathways controlling ERK activation: (i) fast Lck dependent and (ii) relatively slow Lck independent (5,6). We also found that LckY394F can successfully activate ERK with kinetics similar to LckWT, verifying the functionality of LckY394F. The diffusion speed of both LckY394F and LckWT are precisely correlated with their kinase activity in T cells before and after co-receptor activation (Fig. 7). LckY394F diffuses faster at the basal level when it is inactive, while basal LckWT, activated LckY394F, and LckWT all diffuse slower (Fig. 7). Besides diffusion speed as an indicator of the clustering state of a molecule at the plasma membrane, our | 3,068,675 | 195186884 | 0 | 16 |
experiments with Lck-split Venus further confirmed that both LckWT and LckY394F showed an enhanced dimerization and potential clustering after stimulation. These results suggest that the interaction and possibly clustering status of Lck may reflect its kinase function and, in turn, regulate its kinase activities. Rossy et al. have suggested in 2013 an impaired clustering in the Y394F mutant in TCR-activated settings (25). Our study is different in that we used preclustered CD3/CD28 antibodies in the medium, but not coated on cover glass. Therefore, the enhanced Lck-Lck interaction for LckY394F in our study is prominent after activation, in contrast to the previous report in which the antibodies were fixed and (C) Bar graph shows the normalized Venus intensity of LckWT and LckYF in JCam cells before and after activation by CD3/CD28 antibody clusters for 3 hours. N = 3, n = 147, 137, 118, and 136 for LckWT-before, LckWT and CD3/CD28, LckYF-before, and LckYF and CD3/CD28, respectively. Error bars: mean ± SEM; *P < 0.05, ***P < 1 × 10 −3 . (D) Proposed mechanism of Lck interaction and activation. At the resting stage, the LckWT molecules form complexes via phosphorylated Y394 at T cell membrane in a preactivated state. Upon antigen binding, activated LckWT can be further activated and aggregated with the TCR/CD3 complex. The mutant LckYF (Y394F) molecules are relatively unbound and not preactivated, but they can be activated and aggregated similarly to LckWT by CD3/CD28 stimulation. Lck molecules were color coded by their kinase activity, with the cold and hot colors representing low and | 3,068,676 | 195186884 | 0 | 16 |
high activities, respectively. Ab, antibody. immobile on a solid surface. We were also able to visualize Lck kinase activity directly in live cells and connect Lck-Lck interaction with its ability to overcome the deficiency of Y394F to induce kinase activity. It is possible that the phosphorylation of Y394 in LckWT enables a "sticky" property of the molecule, which allows intermolecular interactions, enhancing Lck clustering and enabling its preactivation even at low expression levels in T cells. This hypothesis is also consistent with reports that the open conformation of Lck can be stabilized by intact Y394, which can mediate the clustering of Lck during T cell activation (Fig. 6) (22,25,57), and that LckWT has a higher level of clustering than LckY394F (25). We hypothesized that Lck clustering can affect its kinase activity: Clustered Lck can potentially transphosphorylate a neighboring Lck at Y192 in the SH2 domain (57,58) to reduce the inhibitory binding between the intramolecular Lck SH2 and pY505 (59), leading to the opening and activation of the neighboring Lck, and vice versa. In contrast, LckY394F is unable to cluster at the basal level, and its kinase activity is relatively low and linearly dependent on its expression (Fig. 6). TCR and its costimulation receptor clustering can fully enable the clustering and kinase activation of LckY394F (Fig. 8). As such, we conclude that a major role of Y394 within Lck is to provide a phosphorylatable anchoring motif mediating the Lck-Lck interaction and potentially clustering and maintaining a high basal activity in live cells. In summary, we developed a | 3,068,677 | 195186884 | 0 | 16 |
new substrate-based Lck FRET biosensor with high sensitivity and spatiotemporal resolution to monitor Lck kinase activity in live T cells. Our data showed that the expression of LckWT was strictly controlled in T cells, where the molecules form complexes with slow diffusion and preactivated kinase activity at the basal state. These preactivated Lck population provides a fast and efficient mechanism to phosphorylate and activate TCRs upon stimulation as TCRs contain favorable substrate sequences for Lck (60). On the other hand, although the mutant LckY394F is relatively unbound and less preactivated with fast diffusion rates, CD3/CD28 clustering can trigger the formation of a submembrane compartment to cause the accumulation of CD3/CD28 and LckY394F at the same region for triggering complex formation and achieving activation to the level similar to that of LckWT (Fig. 8). Consequently, both LckWT and LckY394F can mediate fast downstream Lck-dependent ERK activation. These findings highlight the power of our FRET-based ZapLck biosensor in deciphering dynamic Lck regulation and function in TCR signals. Our findings can, hence, provide insight into the functional role played by the Lck kinase in T cell activation and potentially guide the optimization of engineered T cells for therapeutics. DNA constructs and virus production Lck biosensors with different substrates were constructed using a published Src biosensor as the template (31), replacing the Src-specific substrate with Lck-specific substrates. For bacterial expression, the Lck biosensors were inserted into a pRSET-B backbone plasmid and transformed into BL21 competent E. coli cells for protein expression. For the mammalian cell expression, Lck biosensors were inserted | 3,068,678 | 195186884 | 0 | 16 |
into a customized pCAGGS vector using Gibson Assembly (New England BioLabs) (46). The biosensors were transformed into DH5a competent E. coli cells for DNA amplification. LckWT and Lck mutants were inserted into a customized pCAGGS vector. The initial template constructs LckWT-CFP, LckK273R-CFP, and LckY394F-CFP were gifts from Dr. Katharina Gaus (Centre for Vascular Research and Australian Centre for NanoMedicine, University of New South Wales, Sydney, Australia). For lentiviral production, fragments of the ZapLck biosensor, LckWT, LckY394F, and LckK273R, were amplified using polymerase chain reaction (PCR) and then inserted into a pSIN vector between Spe I and Eco RI restriction enzyme sites. For expression level experiments, LckWT and LckY394F were fused with mCherry at the C terminus through overlap PCR and inserted into the pSIN vector for lentiviral production. For visualization and analysis of Lck-Lck interaction, LckWT and LckY394F were fused with VN173 and VC155, respectively, at the C termini and inserted into the pSIN vector through Gibson Assembly. All the fragments were amplified using Q5 high-fidelity DNA polymerase (New England Biolabs, cat#M0491L). The pBiFC-bJunVN173 (cat#22012) and pBiFC-bFosVC155 (cat#22013) used as templates of VN173 and VC155 were purchased from Addgene. To amplify the viruses, human embryonic kidney (HEK) 293T cells were cotransfected with the target plasmids contained in the pSIN vector, pCMVDR, and pCMV-VSVG using Lipofectamine 2000 (Invitrogen) in Dulbecco's minimum essential medium (DMEM) containing 10% fetal bovine serum (FBS). Eight hours later, we changed the medium with advanced DMEM containing 2% FBS and 0.2% penicillin-streptomycin. After 48 hours, we collected the virus and purified the sample | 3,068,679 | 195186884 | 0 | 16 |
with a 0.22-mm filter. To infect Jurkat or JCam cells, the virus was added to cells in a 1:1 volume ratio (with cell concentration below 1 × 10 6 cells/ml). For JCam cells infected with LckWT, LckY394F, or LckK273R, we used puromycin (1 mg/ml) to select positive cells over a duration of 1 week before imaging. In vitro kinase assay The Lck biosensor plasmids were transformed into BL21 competent cells for protein amplification. Biosensor proteins were purified with the Kimble-Chase Protein Purification Kit (Fisher Scientific). For in vitro kinase assays, purified biosensor proteins were diluted to a final concentration of 1 mM in kinase assay buffer [50 mM tris-HCl (pH 8.0), 100 mM NaCl, 10 mM MgCl 2 , 2 mM dithiothreitol] containing 1 mg/ml active Lck, Fyn, or Src kinases in a 96-well plate (100 ml of total volume per well). The reaction was performed at 37°C for 3 hours. The emission spectra of the biosensor solutions were collected every 3 min with the plate reader (Tecan Infinite M1000 PRO using the i-control 1.10 software) before and after the addition of ATP (1 mM). The sample was excited at 427 nm, and the emission ratio (ECFP/FRET) was calculated according to the fluorescence intensity readouts at 476 and 528 nm for ECFP and FRET emission intensity, respectively. Electroporation Cells were passaged the day before electroporation. We washed about 5 to 10 million cells with PBS twice. Opti-MEM medium (500 ml) was added to the resuspended cells. To deliver a single DNA plasmid, we added 15 mg | 3,068,680 | 195186884 | 0 | 16 |
of plasmids to the resuspended cells. To introduce two or three plasmids, we added 10 mg of DNA for each plasmid to the resuspended cells. We then transferred the mixture to the 4-mm cuvette (Bio-Rad) for electroporation (270 V, 950 mF, infinite resistance) while avoiding bubbles. After electroporation, cells were transferred from the cuvette to the 35-mm dish with 5 ml of transfection medium for culture in the incubator. Cell preparation for imaging Jurkat and JCam cells were cultured in RPMI 1640 medium (Gibco) containing 10% FBS, 1% penicillin-streptomycin, 1% sodium pyruvate, and L-glutamine (300 mg/l). HeLa cells and HEK 293T cells were cultured in DMEM (Gibco) containing 10% FBS, 1% penicillin-streptomycin, and L-glutamine (584 mg/l). All cells were incubated in a 5% CO 2 and 95% humidified incubator at 37°C. The cells were introduced with different biosensors and Lck variants via electroporation, transfection, or lentiviral infection depending on the cell types. Specifically, in Fig. 2 and fig. S2, transient transfection or electroporation was used. For Figs. 3 to 7 and figs. S3 to S8, lentiviral infection was used to introduce Lck mutants and Lck biosensors, while ERK biosensors were introduced by electroporation. In general, it is easier to deliver the genes into Jurkat cells and primary human T cells with lentiviral infection. On the basis of our experience, the performance of ratiometric FRET biosensor proteins is similar when expressed via lentiviral infection, lipotransfection, or electroporation. HeLa cells were cultured in DMEM for 36 hours and then transferred onto glass-bottom dishes (Cell E&G) to be starved | 3,068,681 | 195186884 | 0 | 16 |
overnight before imaging. The dishes were coated with fibronectin (10 mg/ml) in PBS at 4°C overnight. During imaging, cells were stimulated with EGF (100 ng/ml). For Jurkat and JCam cells, the cells were seeded onto glass-bottom dishes (Cell E&G) and kept in the incubator for at least 10 min before imaging. In this case, the glass-bottom dishes were precoated with nonspecific immunoglobulin G (IgG) secondary antibody (10 mg/ml). During imaging, the cells were stimulated with CD3/CD28 antibody clusters. The CD3 antibody (10 mg/ml) and CD28 antibody (5 mg/ml) mixture was preclustered with IgG conjugated to biotin and then further coupled with streptavidin, kept at 4°C, before being used to stimulate the Jurkat and JCam cells. This concentration of CD3/CD28 antibody clusters was used as the baseline concentration of 1 unit (CD3 antibody at 10 mg/ml and CD28 antibody at 5 mg/ml), which was then diluted to 1/10 and 1/50 unit to compare the effect of concentration on Lck activation. In specified experiments, SFKs in cells were inhibited by 10 mM PP1 to observe the biosensor responses. Human PBMCs were isolated from buffy coats (San Diego Blood Bank) using Ficoll gradients (Amersham Biosciences). CD3 + T cells were isolated from PBMCs using the Pan T Cell Isolation Kit (Miltenyi). For lentiviral transduction, cells were first activated for 72 hours using PHA (phytohemagglutinin) (Fisher Scientific, R30852801) in complete RPMI medium with IL-2 (100 IU/ml). The cells were then transduced with concentrated ZapLck biosensor lentivirus at a MOI (multiplicity of infection) of 10 by spinoculation on RetroNectin (Takara)-coated plates | 3,068,682 | 195186884 | 0 | 16 |
at 1800g, 32°C, for 1 hour. Microscope imaging and analysis For FRAP experiments, the mCherry images were collected using the MetaFluor 6.2 software (Molecular Devices, Sunnyvale, California) on an epifluorescence microscope (Olympus IX81) with excitation at 572DF35 and emission at 632DF60 using 1% of the light source power. During imaging, the cells were kept in RPMI 1640 medium without serum at 37°C, and the objective was focused at the bottom of the cell. The cells were monitored before photobleaching to confirm that the fluorescence signals were stable during imaging. Photobleaching was conducted by exciting mCherry at 560DF20 in a region of interest with full power of the light source for 15 s. Subsequently, the recovery process was imaged at 5-s intervals for the mCherry fluorescence signals (47). For FRAP analysis, we used a finite element-based diffusion analysis method, which was previously developed and rigorously characterized by us to allow estimation of diffusion coefficients based on images collected by wide-field microscopes. The methods and results have been published in PLOS Computational Biology (47). With this method, the photobleach experiment protocol is simple and does not require the usage of a confocal microscope. The analysis method can be applied to cells of complex geometry without restrictions on the photobleaching pattern and protocol. Only one fluorescence intensity image before photobleaching and two images after are required for estimating the diffusion coefficient without the need of a complete time course. For FRET experiments, the images were collected with a 420DF20 excitation filter, a 450DRLP dichroic mirror, and two emission filters | 3,068,683 | 195186884 | 0 | 16 |
controlled by a filter changer (480DF30 for CFP and 535DF25 for FRET). The pixel-wise images of the CFP/YFP emission ratio were computed to quantify the FRET signals, which represent the concentration of the phosphorylated Lck biosensor and, hence, Lck activity in space and time. A Nikon Eclipse Ti inverted microscope installed with a 300-W Xenon lamp (Atlas Specialty Lighting), an electron multiplying (EM) charge-coupled device camera (QuantEM:512SC, Photometrics), and a 100× Nikon microscope objective (numerical aperture, 1.45) were used to capture all imaging data with the MetaMorph 7.8 software (Molecular Devices). Analysis of all acquired images was conducted on FluoCell, an image analysis software tool developed in the Wang Lab (unpublished results, S.L. at the University of California). Western blot The antibodies used for the immunoblot analysis are as follows: anti-pERK(1/2) (4377T), anti-ERK(1/2) (9102S), and anti-glyceraldehyde phosphate dehydrogenase (2118S) from Cell Signaling Technology. 4G10 was from Millipore (05-321), and anti-GFP (ab290) was from Abcam. Briefly, the cells were stimulated as indicated, and then the cells were lysed using radioimmunoprecipitation assay buffer (9806S) on ice for 10 min and then centrifuged at 4°C and 13,000 rpm. The protein concentration was measured using a Bio-Rad protein assay dye reagent (Bio-Rad, #5000006) following the protocol of the manufacturer. The same amount of boiled proteins was loaded into precast 4 to 20% gradient SDS-polyacrylamide gel electrophoresis (PAGE) (cat#4561096) or 10% precast SDS-PAGE gel for ERK phosphorylation measurement. The gel was run at 110v for 60 to 90 min and then transferred to a nitrocellulose filter membrane at 110v for 60 | 3,068,684 | 195186884 | 0 | 16 |
min. Dry milk (5%) was used to block the membrane at room temperature for 30 min. Then, the membrane was incubated with primary antibody at 4°C overnight, followed by incubation with the secondary antibody at room temperature for 1 hour. Last, the membrane was visualized with standard chemiluminescence. Statistical analysis Two-tailed Student's t tests with unequal variants or the Wilcoxon rank sum tests were used for statistical analysis. The P values were adjusted to the comparison of multiple groups either by the MATLAB multcompare function or by summing the P values from individual tests with a result of significant difference. The analysis methods were also described in the figure legends. N represents the number of independent experiments, and n represents the number of cells. All the results shown in the main figures had three or more independent repeats. SUPPLEMENTARY MATERIALS Supplementary material for this article is available at http://advances.sciencemag.org/cgi/ content/full/5/6/eaau2001/DC1 Fig. S1. Further in vitro characterization of the Lck biosensors. Fig. S2. Characterization of Lck biosensors with different substrates in HeLa cells. Fig. S3. Characterization of the ZapLck biosensor in JCam cells. Fig. S4. Preactivation of Lck in Jurkat and PBMCs. Fig. S5. Characterize the subcellular localization of mCherry-tagged Lck mutants, and the sensitivity of the ZapLck biosensor in detecting Lck activity compared with a Src biosensor. S C I E N C E A D V A N C E S | R E S E A R C H A R T I C L E Fig. S6. Phosphorylation level of endogenous ERK in | 3,068,685 | 195186884 | 0 | 16 |
T cells stimulated by CD3/CD28 antibody clusters. Fig. S7. Activation of Lck kinase in JCam cells with different stimuli at different concentrations. Fig. S8. CD28-only antibody did not trigger TCR signals. Movie S1. The dynamic change of ZapLck biosensor ECFP/FRET ratio signals under CD3/CD28 stimulation and PP1 inhibition in Jurkat or JCam cells with or without Lck or Lck mutants as indicated. Movie S2. The dynamic change of ERK (NES) biosensor FRET/ECFP ratio signals under CD3/ CD28 stimulation and PP1 inhibition in Jurkat or JCam cells with or without Lck or Lck mutants as indicated. | 3,068,686 | 195186884 | 0 | 16 |
Dynamic Multiscaling in Two-dimensional Fluid Turbulence We obtain, by extensive direct numerical simulations, time-dependent and equal-time structure functions for the vorticity, in both quasi-Lagrangian and Eulerian frames, for the direct-cascade regime in two-dimensional fluid turbulence with air-drag-induced friction. We show that differ- ent ways of extracting time scales from these time-dependent structure functions lead to different dynamic-multiscaling exponents, which are related to equal-time multiscaling exponents by different classes of bridge relations; for a representative value of the friction we verify that, given our error bars, these bridge relations hold. The scaling properties of both equal-time and timedependent correlation functions close to a critical point, say in a spin system, have been understood well for nearly four decades [24,25]. By contrast, the development of a similar understanding of the multiscaling properties of equal-time and time-dependent structure functions in the inertial range in fluid turbulence still remains a major challenge for it requires interdisciplinary studies that must use ideas both from nonequilibrium statistical mechanics and turbulence [26][27][28][29][30][31][32][33][34][35]. We develop here a complete characterization of the rich multiscaling properties of time-dependent vorticity structure functions for the direct-cascade regime of two-dimensional (2D) turbulence in fluid films with friction, which we study via a direct numerical simulation (DNS). Such a characterization has not been possible hitherto because it requires very long temporal averaging to obtain good statistics for quasi-Lagrangian structure functions [36], which are considerably more complicated than their conventional, Eulerian counterparts as we show below. Our DNS study yields a variety of interesting results that we summarize informally before providing | 3,068,687 | 30992242 | 0 | 16 |
technical details and precise definitions: (a) We calculate equal-time and timedependent vorticity structure functions in Eulerian and quasi-Lagrangian frames [36]. (b) We then show how to extract an infinite number of different time scales from such time-dependent structure functions. (c) Next we present generalizations of the dynamic-scaling Ansatz, first used in the context of critical phenomena [25] to relate a diverging relaxation time τ to a diverging correlation length ξ via τ ∼ ξ z , where z is the dynamic-scaling exponent. These generalizations yield, in turn, an infinity of dynamic-multiscaling exponents [28,29,[31][32][33][34][35]. (d) A suitable extension of the multifractal formalism [27], which provides a rationalization of the multiscaling of equal-time structure functions in turbulence, yields lin-ear bridge relations between dynamic-multiscaling exponents and their equal-time counterparts [28,29,[31][32][33][34][35]; our study provides numerical evidence in support of such bridge relations. The statistical properties of fully developed, homogeneous, isotropic turbulence are characterized, inter alia, by the equal-time, order-p, longitudinal-velocity struc- is the Eulerian velocity at point x and time t, and r ≡| r |. In the inertial range η d r L, S p (r) ∼ r ζp , where ζ p , η d , and L, are, respectively, the equal-time exponent, the dissipation scale, and the forcing scale. The pioneering work [26] of Kolmogorov (K41) predicts simple scaling with ζ K41 p = p/3 for three-dimensional (3D) homogeneous, isotropic fluid turbulence. However, experiments and numerical simulations show marked deviations from K41 scaling, especially for p ≥ 4, with ζ p a nonlinear, convex function of p; | 3,068,688 | 30992242 | 0 | 16 |
thus, we have multiscaling of equal-time velocity structure functions. To examine dynamic multiscaling, we must obtain the order-p, timedependent structure functions F p (r, t), which we define precisely below, extract from these the time scales τ p (r), and thence the dynamic-multiscaling exponents z p via dynamic-multiscaling Ansätze like τ p (r) ∼ r zp . This task is considerably more complicated than its analog for the determination of the equal-time multiscaling exponents ζ p [28][29][30][31][32][33][34][35] for the following two reasons: (I) In the conventional Eulerian description, the sweeping effect, whereby large eddies drive all smaller ones directly, relates spatial separations r and temporal separations t linearly via the mean-flow velocity, whence we get trivial dynamic scaling with z p = 1, for all p. A quasi-Lagrangian description [28,36] eliminates sweeping effects so we calculate time-dependent, quasi-Lagrangian vorticity structure functions from our DNS. (II) Such time-dependent structure functions, even for a fixed order p, do not collapse onto a scaling function, with a unique, order-p, dynamic exponent. Hence, even for a fixed order p, there is an infinity of dynamic-multiscaling exponents [28,29,[31][32][33][34][35]; roughly speaking, to specify the dynamics of an eddy of a given length scale, we require this infinity of exponents. Statistically steady fluid turbulence is very different in 3D and 2D; the former exhibits a direct cascade of energy whereas the latter shows an inverse cascade of kinetic energy from the energy-injection scale to larger length scales and a direct cascade in which the enstrophy goes towards small length scales [37]; in many | 3,068,689 | 30992242 | 0 | 16 |
physical realizations of 2D turbulence, there is an air-drag-induced friction. In this direct-cascade regime, velocity structure functions show simple scaling but their vorticity counterparts exhibit multiscaling [38][39][40], with exponents that depend on the friction. Time-dependent structure functions have not been studied in 2D fluid turbulence; the elucidation of the dynamic multiscaling of these structure functions, which we present here, is an important step in the systematization of such multiscaling in turbulence. We numerically solve the forced, incompressible, 2D Navier-Stokes (2DNS) equation with air-drag-induced friction, in the vorticity(ω)-stream-function(ψ) representation with periodic boundary conditions: where , and the velocity u ≡ (−∂ y ψ, ∂ x ψ). The coefficient of friction is µ and f is the external force. We work with both Eulerian and quasi-Lagrangian fields. The latter are defined with respect to a Lagrangian particle, which was at the point ξ 0 at time t 0 , and is at the position ξ(t|ξ 0 , t 0 ) at time t, such that dξ(t|ξ where u is the Eulerian velocity. The quasi-Lagrangian velocity field u QL is defined [36] as follows: likewise, we can define the quasi-Lagrangian vorticity field ω QL in terms of the Eulerian ω. To obtain this quasi-Lagrangian field we use an algorithm developed in Ref. [41], described briefly in the Supplementary Material. To integrate the Navier-Stokes equations we use a pseudo-spectral method with the 2/3 rule for the removal of aliasing errors [40] and a second-order Runge-Kutta scheme for time marching with a time step δt = 10 −3 . We force | 3,068,690 | 30992242 | 0 | 16 |
the fluid deterministically on the second shell in Fourier space. And we use µ = 0.1, ν = 10 −5 , and N = 2048 2 collocation points [42] We obtain a turbulent but statistically steady state with a Taylor microscale λ 0.2, Taylor-microscale Reynolds number Re λ 1400, and a box-size eddy-turn-over time τ eddy 8. We remove the effects of transients by discarding data upto time 80τ eddy . We then obtain data for averages of time-dependent structure functions for a duration of time 100τ eddy . The energy spectrum averaged over the same time interval is shown in Fig. (1a). The equal-time, order-p, vorticity structure functions we consider are S φ , the angular brackets denote an average over the nonequilibrium statistically steady state of the turbulent fluid, and the superscript φ is either E, in the Eulerian case, or QL, in the quasi-Lagrangian case; for notational convenience we do not include a subscript ω on S φ p and the multiscaling exponent ζ φ p . We assume isotropy here, but show below how to extract the isotropic parts of S φ p in a DNS. We also use the time-dependent, order-p vorticity structure functions We concentrate on the case t 1 = t 2 = . . . = t l ≡ t and t l+1 = t l+2 = . . . = t p = 0, with l < p, and, for simplicity, denote the resulting timedependent structure function as F φ p (r, t); shell-model studies [31,32] have | 3,068,691 | 30992242 | 0 | 16 |
shown that the index l does not affect dynamic-multiscaling exponents, so we suppress it henceforth. Given F φ p (r, t), it is possible to extract a characteristic time scale τ p (r) in many different ways. These time scales can, in turn, be used to extract the orderp dynamic-multiscaling exponents z p via the dynamicmultiscaling Ansatz τ p (r) ∼ r zp . If we obtain the order-p, degree-M , integral time scale we can use it to extract the integral dynamic-multiscaling exponent z I,φ p,M from the relation T I,φ p,M ∼ r z I,φ p,M . Similarly, from the order-p, degree-M , derivative time scale we obtain the derivative dynamic-multiscaling exponent z D,φ p,M via the relation T D,φ p,M ∼ r z D,φ p,M . Equal-time vorticity structure functions in 2D fluid turbulence with friction exhibit multiscaling in the direct cascade range [38][39][40]. For the case of 3D homogeneous, isotropic fluid turbulence, a generalization of the multifractal model [27], which includes time-dependent velocity structure functions [28,32,34,35], yields linear bridge relations between the dynamic-multiscaling exponents and their equal-time counterparts. For the directcascade regime in our study, we replace velocity structure functions by vorticity structure functions and thus obtain the following bridge relations for time-dependent vorticity structure functions in 2D fluid turbulence with friction: The vorticity field ω φ = ω φ + ω φ can be decomposed into the time-averaged mean flow ω φ and the fluctuations ω φ about it. To obtain good statistics for vorticity structure functions it is important to | 3,068,692 | 30992242 | 0 | 16 |
eliminate any anisotropy in the flow by subtracting out the mean flow from the field. Therefore, we redefine the order-p, equal-time structure function to be S φ p (r c , R) ≡ |ω φ (r c + R) − ω φ (r c )| p , where R has magnitude R and r c is an origin. We next use S φ p (R) ≡ S φ p (r c , R) rc , where the subscript r c denotes an average over the origin (we use r c = (i, j), 2 ≤ i, j ≤ 5). These averaged structure functions are isotropic, to a good approximation for small R, as can be seen from the illustrative pseudocolor plot of S QL 2 (R) in Fig. (1a). The purely isotropic parts of such structure functions can be obtained [40,43] via an integration over the angle θ that R makes with the x axis, i.e., we calculate S φ p (R) ≡ 2π 0 S φ p (R)dθ and thence the equal-time multiscaling exponent ζ φ p , the slopes of the scaling ranges of log-log plots of S φ p (R) versus R. The mean of the local slopes ξ p ≡ d(log S φ p )/d(log R) in the scaling range yields the equal-time exponents; and their standard deviations give the error bars. The equal-time vorticity multiscaling exponents, with 1 ≤ p ≤ 6, are given for Eulerian and quasi-Lagrangian cases in columns 2 and 3, respectively, of Table 1; they are equal, | 3,068,693 | 30992242 | 0 | 16 |
within error bars, as can be seen most easily from their plots versus p in Fig.(1c). We obtain the isotropic part of F φ p (R, t) in a similar manner. Equations (4) and (5) now yield the orderp, degree-M integral and derivative time scales (see the Supplementary Material). Slopes of linear scaling ranges of log-log plots of T I,φ p,M (R) versus R yield the dynamic multiscaling exponent z I,φ p,M . A representative plot for the quasi-Lagrangian case, p = 2 and M = 1, is given in Fig. (1 d); we fit over the range −1.2 < log 10 (r/L) < −0.55 and obtain the local slopes χ p with successive, nonoverlapping sets of 3 points each. The mean values of these slopes yield our dynamic-multiscaling exponents (column 5 in Table 1) and their standard deviations yield the error bars. We calculate the degree-M , order-p derivative time exponents by using a sixth-order, finite-difference scheme to obtain T D,φ p,M and thence the dynamic-multiscaling exponents z D,φ p,M . Our results for the quasi-Lagrangian case with M = 2 are given in column 7 of Table 1. We find, furthermore, that both the integral and derivative bridge relations (6) and (7) hold within our error bars, as shown for the representative values of p and M considered in Table 1 (compare columns 4 and 5 for the integral relation and columns 6 and 7 for the derivative relation). Note also that the values of the integral and the derivative dynamic-multiscaling exponents are markedly | 3,068,694 | 30992242 | 0 | 16 |
different from each other (compare columns 5 and 7 of Table 1). The Eulerian structure functions F E p (R, t) also lead to nontrivial dynamic-multiscaling exponents, which are equal to their quasi-Lagrangian counterparts (see Supplementary Material). The reason for this initially surprising result is that, in 2D turbulence, the friction controls the size of the largest vortices, provides an infra-red cut-off at large length scales, and thus suppresses the sweeping effect. We have demonstrated this in the supplementary material. Had the sweeping effect not been suppressed, we would have obtained trivial dynamic scaling for the Eulerian case. The calculation of dynamic-multiscaling exponents has been limited so far to shell models for 3D, homogeneous, isotropic fluid [29,31,32,34,35] and passive-scalar turbulence [33]. We have presented the first study of such dynamic multiscaling in the direct-cascade regime of 2D fluid turbulence with friction by calculating both quasi-Lagrangian and Eulerian structure functions. Our work brings out clearly the need for an infinity of time scales and associated exponents to characterize such multiscaling; and it verifies, within the accuracy of our numerical calculations, the linear bridge relations (6) and (7) for a representative value of µ. We find that friction also suppresses sweeping effects so, with such friction, even Eulerian vorticity structure functions exhibit dynamic multiscaling with exponents that are consistent with their quasi-Lagrangian counterparts. Experimental studies of Lagrangian quantities in turbulence have been increasing steadily over the past decade [44]. We hope, therefore, that our work will encourage studies of dynamic multiscaling in turbulence. Furthermore, it will be interesting | 3,068,695 | 30992242 | 0 | 16 |
to check whether the time scales considered here can be related to the persistence time scales for 2D turbulence [45]. We thank J. K. Bhattacharjee for discussions, the European Research Council under the Astro-Dyn Research Project No. 227952, National Science Foundation under Grant No. PHY05-51164, CSIR, UGC, and DST (India) for support, and SERC (IISc) for computational resources. PP and RP are members of the International Collaboration for Turbulence Research; RP, PP, and SSR acknowledge support from the COST Action MP0806. Just as we were preparing this study for publication we became aware of a recent preprint [46] on a related study for 3D fluid turbulence. We thank L. Biferale for sharing the preprint of this paper with us. integral above can be evaluated at each time step by an additional call to a fast-Fourier-Transform (FFT) subroutine. The additional computational cost of obtaining u QL at all collocation points is that of following a single Lagrangian particle and an additional FFT at each time step. Numerical determination of integral time scales from time-dependent structure functions To extract the integral time scale, of degree M , from a time-dependent structure function, we have to evaluate the integral in Eq. (4) numerically. In practice, because of poor statistics at long times, we integrate from t = 0 to t = t * , where t * is the time at which F φ p (R , t) = ; we choose = 0.6, but we have checked that our results do not change, within our error bars, for | 3,068,696 | 30992242 | 0 | 16 |
0.5 ≤ ≤ 0.75. This numerical integration is done by using the trapezoidal rule. Dynamic multiscaling for Eulerian structure functions Equal-time Eulerian structure functions have been discussed in our paper above. To obtain time-dependent, Eulerian, vorticity structure functions we proceed as we did in the quasi-Lagrangian case. We obtain the required vorticity increments and from these the purely isotropic part of the time-dependent, order-p structure function F E p (R , t). Equations (4) and (5) now yield the orderp, degree-M integral and derivative Eulerian time scales. For the former we should integrate F E p (R , t) from t = 0 to t = ∞; in practice, because of poor statistics at long times, we integrate from t = 0 to t = t * , where t * is the time at which F E p (R , t) = ; we choose = 0.6, but we have checked that our results do not change, within our error bars, for 0.5 ≤ ≤ 0.75. Slopes of linear scaling ranges of log-log plots of T I,E p,M (R ) versus R yield the dynamic multiscaling exponent z I,E p,1 . A representative plot for the Eulerian case, p = 2, and M = 1 is given in Fig. (2 a); we fit over the range −1.2 < log 10 (r/L) < −0.55 and obtain the local slopes χ p with successive, nonoverlapping sets of 3 points each. The mean values of these slopes yield our dynamic-multiscaling exponents (column 4 in Table II) and their | 3,068,697 | 30992242 | 0 | 16 |
standard deviations yield the error bars. We calculate the degree-M , orderp derivative time exponents by using a sixth-order finite difference scheme to obtain T D,E p,M and thence the dynamic-multiscaling exponents z D,E p,M ; data for the Eulerian case and the representative value M = 2 are given in column 6 of Table II. We find, furthermore, that both the integral and derivative bridge relations, Eq. (6), and Eq. (7). hold within our error bars, as shown for the representative values of p and M considered in Table II (compare columns 3 and 4 for the integral relation and columns 5 and 6 for the derivative relation). The values of the integral and the derivative dynamic-multiscaling exponents are markedly different from each other (compare columns 4 and 6 of Table II) and the plots of these exponents versus p in Fig. (2 b). In Fig. (2 c), we make the same comparison for the quasi-Lagrangian case. Furthermore, a comparison of the quasi-Lagrangian and Eulerian dynamic-multiscaling exponents given in Tables I in the original paper and Table II, respectively, show that these are the same (within our error bars). Demonstration of Infra-red cutoff of the inverse cascade We have shown that in two dimensional turbulence with friction, the Eulerian and the quasi-Lagrangian velocities have the same dynamical exponents. This is because the inverse cascade has a friction-dependent infrared cutoff. To illustrate the development of this cutoff scale, we have carried out DNS studies of 2D fluid turbulence with µ = 0.01, 0.05, and 0.1, 1024 | 3,068,698 | 30992242 | 0 | 16 |
2 collocation points, and forcing at a wave-vector magnitude k = 80; our DNS studies resolve the inverse-cascade regime in the statistically steady state. The energy spectra from these DNS studies, plotted in Fig. (2d), show clearly that, as µ increases, the inverse cascade is cut off at ever larger values of k. Thus, the friction produces a regularization of the flow and suppresses infrared (sweeping) divergences. | 3,068,699 | 30992242 | 0 | 16 |
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