microalgae - based processes 
 are considered one of the most promising 
 alternative technologies for the production of liquid fuels in the 
 transport sector . the main advantages 
 of microalgae with respect to other possible feedstock are the high 
 potential productivity and the absence of competition with traditional 
 crops for arable land and clean water . however , this potential is 
 still theoretical , and algae production on large scale is not profitable 
 yet . several issues need to be addressed to reach this objective , 
 ranging from algae cultivation and harvesting as well as products 
 extraction . the availability of reliable models 
 would be greatly beneficial 
 as the possibility to represent the fundamental physical , chemical , 
 and biological phenomena would allow one to assess the interactions 
 between equipment design and product yields and to scale - up and optimize 
 the process design and operation . however , to be reliable a predictive model requires a specific 
 set of parameters to be estimated in the most precise and accurate 
 way . one important advantage of a model whose parameters have been 
 reliably identified is that it can be used to predict the system response 
 also in conditions ( significantly ) different from those tested during 
 the identification experiments , which is the typical case when a model 
 is exploited for process scale - up and optimization . to achieve that , 
 however , model parameters must be identifiable , and this may present 
 critical issues even in relatively simple models . in this work we will discuss a modeling structure 
 to represent 
 some key phenomena in algae growth by giving special attention to 
 the identifiability of the proposed model . algae growth is affected 
 by several variables such as nutrient availability , temperature , and 
 mixing , etc . however , being algae photosynthetic organisms , light 
 is the key variable determining growth efficiency and kinetics : for 
 this reason , the focus of this work will be on the representation 
 of its influence on growth . it is worth stressing that the correlation 
 between light intensity and growth is a very complex one , and while 
 low irradiation is limiting , its excess drives to the formation of 
 reactive oxygen species and has an inhibitory effect . in the literature it is possible to divide such models into two 
 main groups :  physiological models  and  state 
 models  . physiological models attempt to describe the dynamic 
 behavior of photosynthetic cells and propose approximations for the 
 actual mechanisms involved in the cells growth . these models 
 may try to represent the optimal allocation of energy and nutrients 
 during cells activities ( e.g. , ross and geider ) or to represent a specific metabolic reaction ( e.g. , marshall 
 et al . , where the damage and repair cycle 
 of protein d1 is described ) . usually these models are extremely detailed 
 and involve a large amount of variables and parameters . the actual 
 identification procedure may be extremely complex ( sometimes even 
 impossible ) and require numerous , highly specific , and costly experiments . state models are instead based on the concept of a photosynthetic 
 unit ( psu ) and are more instrumental for simulating and optimizing 
 industrial cultivation systems . the psu is defined as the sum of the 
 light harvesting complex , the reaction center , and the associated 
 apparatus , which are activated by a given amount of light energy to 
 produce a certain amount of photoproduct . these models are called state models because the psu can be in different 
 states of excitation . one of the first proposed state models was the 
 model by fasham and platt , whose objective 
 was to describe photoinhibition . we will consider the models by camacho rubio 
 et al . and by eilers and peeters ( initially 
 developed 
 by eilers and peeters and then improved 
 by wu and merchuk ) . these models are 
 capable of representing the key phenomena of interest in this work , 
 and they are reasonably simple so as to limit possible identifiability 
 issues . in the eilers and peeters model , the authors assume that if 
 an activated psu absorbs an additional photon , it may become inhibited . 
 for this reason they assume the rate of photoihibition to be proportional 
 to light intensity . it is also assumed that photosynthesis ( and by 
 consequence biomass growth ) is proportional to the transition between 
 the activated state and the resting state . later wu and merchuck modify the model of eilers and peeters , introducing 
 a constant maintenance factor in the description of biomass growth . in the model by camacho rubio et al . both photoinhibition and photoacclimation , which represent the cells 
 response to optimize their photosynthetic apparatus to different light 
 intensities , are considered ( note that in the original work , and in 
 several others , photoacclimation is instead called photoadaptation : 
 photoacclimation is however a more accurate definition : in fact , adaptation 
 refers to the organisms modification during evolution to their 
 environment , thus responses with time scales extremely longer that 
 the ones considered here ) . photoacclimation was at first represented 
 as a steady state process but more recently extended by the same authors 
 in order to represent its dynamics , together with the effect of nonphotochemical 
 quenching ( npq ) and dark respiration ( camacho rubio et al . ) . this last model is indeed very flexible , but 
 the number of model parameters to be estimated is very high and their 
 precise identification may become a long and difficult task , especially 
 if a limited amount of data is available , and for this reason it will 
 not be discussed further in this work . for model development 
 and identification we consider experimental 
 data referring to a particular species of microalgae of industrial 
 interest ( nannochloropsis salina ( n. salina ) ) grown in nonlimiting nutrients conditions and in a flat - plate 
 photobioreactor . these data sets were 
 selected because experimental conditions were optimized to minimize 
 all influences on algae growth other than light intensity . in fact , 
 nutrients and co2 were provided in excess , but also the 
 photobioreactor light path was minimized to reduce as much as possible 
 light attenuation due to cells shading and scattering . sforza 
 et al . demonstrated that this assumption 
 was an acceptable approximation , especially considering that we are 
 interested in representing the exponential growth phase , where nutrient 
 availability is high and cell concentration low . accordingly , these 
 data represent an accurate description of the influence of light alone 
 on algae growth , minimizing the effect of other parameters . it is 
 worth clarifying that other phenomena , which also play a major influence 
 on algae productivity in industrial photobioreactors , such as the 
 dark / light cycles due to mixing are not considered in this model . on the methodological 
 side , a step by step approach will be presented and applied to guarantee 
 the identifiability of the growth model . second , the utilization of 
 both biomass concentration ( growth curves ) and multiple fluorescence 
 measurements will allow shading light on some fundamental phenomena 
 in the correlation between illumination and growth ; in particular , 
 differently from previous contributions , measurements of the light 
 profile of psu saturation will be exploited . the work is organized 
 as follows : after outlining the general identification 
 methodology , the available experimental data and the two candidate 
 models will be introduced and discussed . the successive section is 
 about model discrimination and the enhancement of the selected model . then , an identifiability analysis and a reparameterization approach 
 will allow setting up an identifiable model . identifiability is a key issue to guarantee reliability 
 and predictive 
 capability in a model being developed . figure 1 outlines the basic tasks and information flux required to achieve 
 such a target . the preliminary step is to identify the phenomena that 
 need describing and the fundamental mathematical laws that should 
 be implemented to represent them . in other cases , preliminary experimental 
 data may be needed to envisage the correlation among data and to set 
 up a suitable physical interpretation through a mathematical model . typically , 
 some available data may be exploited at this stage to choose among 
 competitive modeling approaches through suitable discrimination techniques . at least in the easiest cases a  test on experimental 
 data may be sufficient to make the discrimination.ad hoc experiments can 
 also specifically be designed to allow for a more effective and reliable 
 discrimination among different candidates . once a suitable 
 candidate model has been selected ( and a preliminary estimation of 
 its parameter has been carried out ) , the model may need upgrading 
 to improve its capability of representing the phenomena being investigated 
 ( new experiments may be needed and possibly designed , and an estimation 
 of all model parameters should be attempted ) . then it is extremely important to verify the model identifiability , 
 i.e. , to confirm that the optimal set of parameters values is unique 
 and that their values can be determined in a precise way ( and ideally 
 in a physically meaningful way ) . if some identifiability issues arise , 
 then a first approach to tackle the problem is to reparameterize the 
 model . once the model is proved 
 to be identifiable , the final parameter 
 estimation can be performed . note that even when a model is identifiable , 
 measurements of noise and other uncertainty effects may still hinder 
 its practical identifiability , although properly designed experiments 
 may help in tackling the issue . once 
 the final parameters estimation has been performed , new data , not 
 involved in model calibration , should be used to validate the model . this approach has been applied to the specific case study and is discussed 
 in the following sections . for simulation and parameter estimation 
 purposes , the main advantage of 
 state models is that they reduce the complexity 
 of photosynthesis into a few possible states of the psus . this simple 
 structure is also particularly effective in the use of fluorescence 
 measurements , which can be exploited to monitor the psu populations 
 at different states . a psu is defined as the sum of the light harvesting 
 complex , the reaction center , and the associated apparatus , which 
 are activated by a given amount of light energy to produce certain 
 quantities of photoproducts . the eilers and peeters model in the form 
 proposed by wu and merchuck ( afterward 
 denoted as epm ) and the camacho rubio model ( camacho rubio et al . , later called crm ) are two of the simplest models 
 that can describe photosynthetic biomass growth as a function of light 
 intensity . both models consider that a psu can assume three different 
 states of excitation : ( 1 ) the resting ( or open ) state , which is the 
 state of psu before the light energy excites the reaction center ; 
 ( 2 ) the activated ( or closed ) state , that is the state of psu excited 
 by light energy ; ( 3 ) the inhibited state , which is the state of psu 
 damaged by an excess of light energy . both models do not consider 
 any limitation on nutrients availability or mass transport of nutrients : 
 i.e. , only the exponential growth phase is described . this is a reasonable assumption also in our case study , since acclimation 
 characteristic time scale ( hours or days ) is significantly larger 
 than the time scales of the other phenomena being investigated . in the work of wu and merchuck epm 
 was used to fit the data of experiments carried out in a thin tubular 
 loop reactor . part of the reactor was kept in dark to simulate mixing ; 
 light intensities used for the experiments were 110 , 220 , and 550 
 e/(m s ) . the measurements used to calibrate the 
 model were both biomass concentration and dark fluorescence measurements 
 ( fluorescence measurements will be described in section 4 ) . each experiment was carried out for 48 h , and measurements 
 were taken every 12 h. in the work of camacho rubio et al . , crm was applied to a wider range of light intensities 
 ( ranging from 0 to 2000 e/(m s ) ) and to different 
 light regimes ( constant light , flashing light , and day  night 
 cycle ) . data used by the authors were growth rate constant and p  i curves taken from the literature . epm assumes that the number of psus is constant with respect to 
 light intensity and accordingly refers to the psu x1 , x2 , and x3 to represent the resting , activated , and inhibited states , 
 respectively . conversely , crm assumes that the number of psus is a 
 function of light intensity ( indicated as at ) , and the model equations are expressed as a function of the amount 
 of the psus in each of the three states ( a1 , a2 , and a3 ) . x1 , x2 , and x3 represent 
 the fractions 
 of psus in resting , activated , and inhibited states , respectively . a1 , a2 , and a3 are the numbers 
 of psus in resting , activated , and inhibited states , respectively ; 
 at represents the total number of psus . in figure 2a , the three psu states are represented by circles 
 and the possible state transitions are represented by the arrows . the resting state psu can capture light energy and transfer it to 
 an activated state . the psu in the activated state can be damaged 
 by light , or pass down the energy to start the dark phase of photosynthesis 
 ( and then return to a resting state ) . the reaction rate of the transitions 
 involving the absorption of light ( i.e. , x1  x2 and x2  x3 ) is assumed to be 
 first order with respect to light intensity . the other two transitions 
 are assumed to be zero order with respect to light intensity . each 
 transition is assumed to be first order with respect to the psu fraction 
 involved in the transition . the growth rate constant (  ( h ) ) is assumed to be proportional to 
 the state transition from activated to resting state , representing 
 the photochemical reactions . considering that the growth rate can 
 be negative in the dark or at very low light intensity , a constant 
 maintenance ( m ( h ) ) factor is introduced . the model equations are as follows:1234the set of parameters ( whose physical meaning 
 is summarized in table 1 ) is represented by 
 vector  = [ kaep , kdep , kiep , krep , kpep , m ] . in crm , the photoinhibition 
 rate is assumed to be proportional to the sum of resting state and 
 activated psus , i.e. , the active psus represented by a1 + a2 [ psus / cells ] in figure 2b . as in epm , psu activation reaction is assumed 
 to be first order with respect to light intensity and to the amount 
 of resting state psus . however , in crm the transition from activated 
 to resting state , related to biomass growth , is defined as a michaelis menten 
 kinetic , assuming an enzymatic reaction as the limiting step of this 
 process . also , from an analysis of experimental data , the authors assume that the photoinhibition reaction rate 
 is first order with respect to the square root of light intensity . 
 as in epm the recovery of damaged psus is assumed to be a first order 
 reaction with respect to the number of damaged psus . finally , as anticipated , 
 crm includes photoacclimation . the total amount of psus ( at [ psus / cells ] ) in crm as for epm , the growth rate 
 constant (  ( h ) ) is assumed 
 to be proportional to the transition from activated to resting state 
 and a constant maintenance factor ( m ( h ) ) is introduced . the model equations are 
 as follows:56789as reported in the literature , it appears 
 that the condition kscr  a2 is 
 true for the entire range of light intensities considered in the experiments 
 ( ranging from 50 to 1000 e/(m s ) ) . menten 
 kinetic may be well approximated by a first order kinetic as in epm . accordingly , the reduced set of parameters is represented by vector 
  = { kacr , kdcr , kicr , krcr , kpcr , kccr , m } ( parameters 
 physical meanings and units are reported in table 2 ) . the aim of this work is to describe 
 the growth of microalgae in 
 nonlimiting nutrient conditions and according to the hypothesis that 
 the light intensity is constant with respect to the culture time and 
 depth . the fundamental phenomena to be described are as follows : ( i ) 
 reaction centers oxidation / reduction cycle , to represent the photosynthesis , 
 and ( ii ) the damaging effect of excess light on psus ( photoinhibition ) . , microalgae were acclimated 
 to the light used and to the geometry of the photobioreactor . each 
 experiment was conducted in parallel at least twice and in two identical 
 photobioreactors , in order to ensure its reproducibility . only the data in the exponential phase of the 
 original growth curves were used for the parameters estimation , since 
 our model represents only exponential growth and does not consider 
 nutrients limitation . experimental setup was built to limit as much 
 as possible cells shading by decreasing as much as possible the light 
 path and working at low cells concentration . this , together with 
 the presence of nutrients and co2 in nonlimiting amounts , 
 ensures that growth is dependent only from the light intensity reaching 
 the culture . moreover , experimental measurements of fluorescence 
 will be used 
 as additional data for the parameters estimation . these data are commonly available for the photosynthetic 
 organisms and have been exploited to estimate photosynthetic efficiency 
 in a large body of experimental literature ( reviewed by maxwell and 
 johnson ) . in our case we considered in 
 particular two fluorescence parameters , related to the oxidation state 
 of the algal cells : the parameters q ( or fv / fm ) , and ql , whose physical meanings will 
 be briefly detailed in the following . parameter q = fv / fm ( with fv = fm  f0 ) was 
 measured for each light intensity at which microalgae were grown . conversely , the maximum fm is the fluorescence after a saturating light flash , 
 when all reaction centers are closed . here the fluorescence is higher 
 than f0 because saturated psus are unable 
 to perform photochemistry and therefore a larger amount of excitation 
 energy is re - emitted as fluorescence . when psus are active , there 
 is a large difference between fm and f0 ( and thus a larger fv / fm ) , while if psus are photoinhibited , fm is closer to f0 . for this reason , the fv / fm parameter is commonly used in the literature to quantify 
 active psus in vivo . the precise value of fv / fm in the case 
 of fully active psus is known to be variable between species , since 
 it depends on specific properties such as the antenna size . here it 
 is set to be equal to 0.65 as this is a value commonly measured in 
 several microalgae and also in healthy nannochloropsis cultures , exposed to low light . in order to have a parameter bound between 0 and 1 , it is possible 
 to define the parameter qnorm = q / qmax , where qmax is the value of fv / fm in the case of fully active psus . in 
 cultures exposed to different light intensities , a decrease 
 in this value indicates the presence of photoinhibited psus , as normally 
 experienced in high light conditions . for this reason , it can be used 
 to estimate the content of photoinhibited psus and thus it can be 
 correlated to x3 and a3 populations , using the definition of epm and crm , respectively . while q is rather commonly employed in several 
 similar studies , in order to have a better 
 representation of the oxidation state of the psus in illuminated cells , 
 here we also included the fluorescence parameter ql , which provides a linear estimation of the saturation 
 level of psu as discussed in detail in the work by kramer et al . this means that ql is 1 when all active psus are open , while it decreases to 0 when 
 all psus are closed and photosynthesis is saturated . parameter ql = [ ( fm  fs)/(fm  f0)](f0/fs ) was measured 
 for 21 different light intensities with a pam fluorometer . here fm and f0 stands for the same minimal and maximal fluorescence , 
 measured in light adapted cells , while fs stands 
 for stationary fluorescence in illuminated cells . thus , ql can be exploited as an estimation of the relative ratio 
 of x1 and x2 populations ( a1 and a2 according to crm ) . the complete set of experimental 
 measurements used in this work is reported in appendix 
 a. the first objective is to discriminate between 
 the two alternative 
 models so as to select the most suitable one to describe our system . parameter estimations were performed for both models , based on the 
 entire set of experimental data . the different performance is quantitatively 
 summarized by the  test ( in the case of epm , we 
 have  = 890.1 , whereas , for crm ,  = 301.6 ) . figure 3 shows the behavior 
 of the two models 
 in representing fluorescence experimental profiles . although both 
 fittings are quite unsatisfactory , crm clearly outperforms epm . in 
 fact , figure 3a shows that although epm provides 
 a slightly better fit for the profile of q at low 
 light intensities , only crm is capable of representing the trend at 
 a higher intensity . figure 3b , however , shows 
 that although epm performance is still the worst one , both models 
 can not properly represent the oxidative state of psu in light adapted 
 cells , consistent with the fact that these kinds of measurements were 
 not considered in building such models . this means that both models 
 are not accurate in estimating the psu oxidative state . measurement 
 ( black circles ) and predicted values of ( a ) qnorm and ( b ) ql . 
 red solid lines represent the profiles according to epm , while the 
 dashed green lines represent the profiles according to crm . also in the predictions of the 
 growth rate constant , reported in 
 figure 4 , crm outperforms epm . in fact , epm 
 does not predict the correct value of light intensity at which the 
 maximum growth rate is reached and underestimates the growth rate 
 constant at low light . crm correctly predicts the optimal light intensity 
 but it overestimates the growth rate constant at low light . no additional experiments are needed for discrimination purposes 
 and crm is then selected , although the experimental data demonstrate 
 that further improvements are required to improve the fitting in the 
 region between 150 and 700 e/(m s ) with respect 
 to the psu oxidation state . it is worth underlining that the main 
 difference between epm and crm is the presence , in the latter , of 
 a photoacclimation term expressed through the number of psus per cell . this suggests that the inclusion of this response is absolutely necessary 
 for an accurate description of the photosynthetic performances and 
 is consistent with its biological relevance , demonstrated by the fact 
 that acclimation responses are conserved in all photosynthetic organisms . growth 
 rate constant predicted by the ep ( solid line ) and cr models 
 ( dashed line ) and experimental values of the growth rate constant 
 ( black squares ) . to improve the model , 
 a more detailed description of some fundamental biological phenomena 
 needs to be introduced . according to several works in the literature , therefore , we assumed that photoinhibition 
 does not depend on the number of active psus , but is simply related 
 to light intensity . above icr ( the light 
 intensity where photosynthesis is saturated ) a second process ( photoprotection ) is activated : in these conditions a fraction of the energy absorbed 
 does not lead either to photochemistry or to psu damage but is simply 
 dissipated ( e.g. , because of npq ) . accordingly , eq 6 has been modified to represent the different behavior above 
 and below icr . furthermore , in eq 6 , the reaction order with respect to the light intensity 
 is assumed to be 0.5 : since there is no clear physical reason for 
 setting such a value , here we decided to increase the model flexibility 
 and its capability of incorporating all energy dissipation phenomena , 
 by making the reaction order a parameter (  ) to be estimated . as mentioned above , photoprotection mechanisms are activated when 
 photochemical reactions are saturated , and accordingly parameter icr plays a key role in representing this behavior . the value of critical light intensity when photosynthesis is saturated 
 has been fixed to 150 e/(m s ) , which is the light 
 intensity in response to which nannochloropsis growth 
 is maximal , and the limit over which the growth rate is not linearly 
 dependent on light anymore . this choice 
 is well consistent with the observation of npq dependence from light 
 intensity observed in nannochloropsis cells , which 
 shows activation only over this limit ( npq data are reported in appendix b ) . this parameter thus depends on both 
 light intensity and the number of active psus . also eq 8 representing photoacclimation is 
 modified to allow for a higher flexibility : the hyperbolic form is 
 retained , but the light exponent becomes an additional parameter ( 2 ) to be estimated ( see later on , eq 13 ) . finally , the representation of the maintenance factor is 
 modified , 
 too . in crm the maintenance factor is treated as a constant ( in fact , 
 this is a typical assumption ) . however , the maintenance factor should 
 vary with light intensity to account for the metabolic cost of repairing 
 damaged psus . in order to do this , the 
 easiest way is to express the maintenance factor as a linear function 
 of the damaged psus fraction . since the fluorescence measurements 
 return the number of active psus , the variable term of the maintenance 
 factor was related to the difference between the maximum value of 
 fluorescence ( qmax ) and the current value 
 of fluorescence ( q ) . finally the fluorescence measurements q and ql are related to the 
 oxidation state of the psus , as discussed in section 4 . the modified crm ( mcrm ) is thus constituted by the 
 following set 
 of equations:1011121314151617where a new vector of the model parameters 
 is  = [ ka , kd , ki,0,ki,1,kr , kp , kc , km , m0,,2 ] . in order to be reliable and 
 suitable for process simulation and 
 optimization , the model for photosynthetic biomass growth has to be 
 identified against the experimental data . it can be verified that 
 mcrm shows identifiability issues if a parameter estimation is performed 
 on the whole parameters vector . the estimation of the 
 model parameters is characterized by large confidence intervals for 
 some parameters , and more than one set of optimal values can be determined 
 ( i.e. , the model is not uniquely identifiable ) . in order to overcome 
 this problem both global ( or structural ) and practical identifiability 
 of the model first of all , the global identifiability 
 has been verified using a differential algebra based method . afterward , 
 the practical identifiability has been assessed through a sensitivity 
 analysis and a model reparameterization has been performed . the 
 first step for testing model identifiability is represented by global 
 identifiability analysis . global identifiability analysis is in fact 
 a necessary condition for practical identifiability and can provide 
 the minimum number of observations required to identify an unique 
 set of optimal parameter values . the two hypotheses , upon which structural 
 identifiability analysis rely , are as follows : ( i ) complete absence 
 of measurement errors and those two assumptions refer to an ideal case , and therefore , once 
 a model is verified to be globally identifiable , practical identifiability , 
 too , needs assessing . several methods to study 
 the structural identifiability of nonlinear models have been developed 
 in the past two decades and are nicely reviewed in the work by miao 
 et al . a rather common approach 
 is given by the series expansion method ( e.g. , dochain et al . it requires that functions representing the model 
 are infinitely differentiable , as the method involves the calculation 
 of arbitrary order derivatives . however , the series expansion method has a serious drawback : 
 for high dimensional models , high order derivatives are necessary 
 and the resulting equations can easily become too complicated to solve . in fact , it was verified that in this case the series expansion method 
 leads to an intractable problem . thus , in order to overcome the difficulties 
 related to high order derivatives calculation and the resolution of 
 the resulting equations , a method based on differential algebra was 
 taken into account . in particular , the software package daisy has been used here . daisy , as with all differential 
 algebra based methods , requires equations of the models to be written 
 in the form of differential polynomials . for this reason , mcrm has been slightly modified as detailed in appendix c. results indicate that a single 
 experiment is not sufficient to 
 identify a unique value of all of the parameters . however , if ( at 
 least ) three parallel experiments are carried out at different light 
 intensities and both concentration and fluorescence parameters are 
 measured during the experiments , the model is globally identifiable . note that in 
 several cases , where more complex models need considering , the analysis 
 of the model practical identifiability may be the only test that can 
 be carried out , since the verification of global identifiability may 
 not be viable . as the model is structurally identifiable , its practical 
 identifiability the sensitivity qik of 
 the ith response to the kth model 
 parameter is defined as18where yi is the ith measured responses predicted by 
 the model , y is the same response obtained 
 from a perturbed value of the kth parameter k , and k is the perturbation ( nm is the 
 number of measured responses and n is the number of model parameters ) . the principal goal of sensitivity 
 analysis is to evaluate the impact of each parameter on the measured 
 responses and to underline the presence of correlation among specific 
 subsets of model parameters . in an overparameterized model , near - zero 
 sensitivity values would be obtained , leading to the nonidentifiability 
 of some subsets of model parameters . in the most desirable case , sensitivity 
 profiles should be clearly distinct and far from being symmetrical 
 ( i.e. , they should present a low mutual correlation ) . as the sensitivity 
 analysis requires prespecified parameter values , values by a preliminary 
 parameter estimation have been considered . although not shown here for the sake of conciseness , 
 it was verified that all sensitivity profiles behave as exponential 
 curves . as a consequence , the final value of sensitivities ( value 
 at 120 h ) is sufficient to analyze the system behavior in different 
 experimental conditions . the sensitivity profiles evaluated at four 
 different light intensities are reported in figure 5 . for light intensities under the critical value , the 
 sensitivities 
 related to parameters ki,0 and kr show opposite values . above the critical value 
 of light intensity , the sensitivity of kr is the opposite of the sum of the sensitivities of ki,0 and ki,1 . this suggests 
 that it may be difficult to exploit biomass concentration measurements 
 to identify ki,0 and kr . another critical aspect is related to the fact 
 that parameters ka and kp exhibit 
 a very similar sensitivity at low light conditions ( availability of 
 growth data at high light intensities may be necessary for a robust 
 estimation of these parameters ) . also it should be noticed that the 
 parameter related to photoacclimation , kc , and the parameter representing the maintenance factor in the dark , m0 , exhibit a very similar ( and low ) sensitivity 
 in all light conditions . final values of dynamic sensitivities for mcrm , 
 evaluated at different 
 light intensities . fluorescence profiles 
 ( q and ql ) are also considered . 
 because the dynamics of psus are 
 fast with respect to the sampling time , and our measurements are steady 
 state measurements , only the steady state values of sensitivities will be reported . in figure 6a , b the values 
 of the sensitivities of q and ql are reported for four different light conditions . the sensitivities 
 of parameters 2 , kc , kp , km , and m0 are not reported , since they are zero for 
 both fluorescence measurements ( they are related to biomass growth 
 and are not concerned with the oxidation state of the psus ) . final values 
 for dynamic sensitivities for the camacho rubio model 
 evaluated at different light intensities considering literature values 
 for model parameters . considering the steady state sensitivity values of the fluorescence 
 measurements , the following critical aspects can be noticed : ( 1 ) parameters ka and kd do not 
 affect the dark fluorescence measurements , while they are completely 
 anticorrelated if light fluorescence measurements are available , showing 
 opposite sensitivities in all light conditions ; ( 2 ) parameters ki,0 , ki,1 , kr , and  are not affected by light fluorescence 
 measurements ; ( 3 ) parameter kr is anticorrelated 
 with ki,0 , under the critical value of 
 light intensity , and with ki,1 , above 
 the critical value of light intensity ; ( 4 ) the sensitivity of ki,0 is always quite small . the results 
 suggest that a model reparameterization may be necessary 
 to help tackling the issue . because the fluorescence 
 data are measurements of the psu oxidation state under steady state 
 conditions , an analytical expression for fluorescence measurements 
 can be derived from eqs 10 to 12:1920with r0 = ki,0/kr , r1 = ki,1/kr , and r2 = ka / kd . the interesting 
 aspect is that only four parameters affect the steady state values 
 of the fluorescence measurements : the three ratios r0 , r1 , and r2 and parameter . this suggests that , in order 
 to have a practically identifiable model , the values of two parameters 
 affecting the psu dynamics have to be fixed if no dynamics in the fluorescence 
 data can be included in the data set . it was chosen to fix the values 
 of kr and kd , as those parameters represent the rate constant of the recovery 
 and deexcitation processes , respectively . an approximated estimation 
 can be obtained , considering the time scales of the processes involved : 
 according to literature , values of 100 
 s for ka and of 2.22 
  10 s for kr were assumed . the very low sensitivity to available 
 measurements and the high correlation of kc and m0 make it impossible to identify 
 the two parameters . however , in the case of m0 , previous experiments suggest it to be between 5% and 10% 
 of the maximum growth rate . here we set m0 = 1.5  10 h ( 8% of the maximum growth rate ) . for parameter kc the preliminary estimation suggests that a 
  small  value is required for a good description of 
 data . since in eq 13 we have that i  1 , we verified that kc = 1 is a good approximation for representing 
 all experimental conditions . after reparameterization , the vector 
 of parameters to estimate is  * = [ r0,r1,r2,kp , km,,2 ] . in the first part of this section the mcrm 
 parameter estimation 
 results will be presented and discussed . in the second part two additional 
 growth curves and two new measurements of dark fluorescence will be 
 used to validate the model . in the estimation 
 procedures , parameters are normalized with respect to the initial 
 values obtained by the preliminary parameter estimation , to increase 
 numerical robustness . the results of parameters estimation are reported 
 in table 3 , along with the confidence intervals 
 and the t - values ( for a statistically precise estimation 
 of a model parameter the t - value has to be greater 
 than a reference t - value ) . the t - value statistic shows that the parameter values are estimated in 
 a statistically satisfactory way . the profiles of fluorescence , predicted after the 
 identification 
 of the reparameterized mcrm , are reported in figure 7 . we can observe that the model correctly fits both the dark 
 fluorescence profile ( figure 7a ) and the light 
 fluorescence measurements ( figure 7b ) . biomass 
 growth profiles are rather well represented by the model as illustrated 
 in figure 8 , where the six different illuminating 
 conditions are represented . measurement ( black circles ) and predicted values 
 of ( a ) q and ( b ) ql . 
 blue solid lines 
 represent the profiles predicted by the modified camacho rubio model . biomass concentration 
 profiles at different light intensities predicted 
 by the modified camacho rubio model . predicted 
 by the model is reported along with the experimental value ( i.e. , 
 the value obtained fitting the experimental data of growth during 
 the exponential phase with an exponential curve ) . we can observe that , 
 for all light intensities at which an experiment was carried out , 
 the model well describes the experimental values of the growth rate 
 constant . this suggests that , thanks to the fundamental input of the 
 fluorescence parameters in illuminated cells ( ql ) , the model is capable of reproducing with sufficient accuracy 
 the basic processes of photosynthesis : photochemistry , light damage , 
 and also energy dissipation . growth rate constant predicted by the modified 
 camacho rubio model 
 ( solid line ) and experimental values of the growth rate constant ( black 
 circles ) . red stars represent the experimental values of the growth 
 rate constant for the experiments used in the model validation . from a statistical point of view , 
 the predicted profiles have a 
  value of 117.7 ( whereas  = 
 890.1 in the case of epm and  = 301.6 for crm ) . in order to validate 
 the model , two experiments , not used for model calibration , will be 
 taken into account . in particular two growth curves at 350 and 750 
 e/(m s ) have been considered . for both cultures 
 also the value of q has been measured and was exploited 
 for model validation . in figures 7a and 9 the validation points are represented by red stars . 
 in figure 10 the growth curves have been reported . biomass 
 concentration profiles at different light intensities predicted 
 by the modified camacho rubio model . we can observe that the predictions are accurate for the 
 parameter q. also the growth curves are predicted 
 in a sufficiently 
 accurate way , although the growth at 350 e/(m s ) 
 is slightly overestimated by the model , while the growth at 750 e/(m s ) is somewhat underestimated in the final part . from a statistical 
 point of view a literature 
 model was selected and 
 modified to describe microalgae growth through a rigorous identification 
 procedure . the estimation of the model parameters was performed considering 
 experimental data ( growth profiles and fluorescence measurements ) 
 on n. salina . results show that the developed model 
 well represents biomass growth over a wide range of light intensities . the modified model was also able to reproduce fluorescence measurements , 
 including the light profile of psu saturation . this suggests that 
 the proposed model is accurate enough to represent all major processes 
 of photosynthesis , photochemistry , psu damage , and energy dissipation . while results in reproducing experimental data are fully satisfactory , 
 it should be underlined that algae growing in an industrial scale 
 photobioreactor are exposed to different conditions . in particular 
 light is not homogenously distributed because of cells shading , and 
 illumination intensity is not constant because of diurnal changes 
 and cells mixing . future efforts will be made to expand the model to include these phenomena , 
 also by designing appropriate experiments .
despite the high potential as feedstock 
 for the production of fuels 
 and chemicals , the industrial cultivation of microalgae still exhibits 
 many issues . 
 yield in microalgae cultivation systems is limited by 
 the solar energy that can be harvested . 
 the availability of reliable 
 models representing key phenomena affecting algae growth may help 
 designing and optimizing effective production systems at an industrial 
 level . in this work the complex influence of different light regimes 
 on seawater alga nannochloropsis salina growth 
 is 
 represented by first principles models . 
 experimental data such as in vivo fluorescence measurements are employed to develop 
 the model . 
 the proposed model allows description of all growth curves 
 and fluorescence data in a reliable way . 
 the model structure is assessed 
 and modified in order to guarantee the model identifiability and the 
 estimation of its parametric set in a robust and reliable way .