Model Discrimination of Polynomial Systems via Stochastic Inputs

Transcription

1 Model Discrimination of Polynomial Systems via Stochastic Inpts D. Georgiev and E. Klavins Abstract Systems biologists are often faced with competing models for a given experimental system. Unfortnately, performing experiments can be time-consming and expensive. Therefore, a method for designing experiments that, with high probability, discriminate between competing models is desired. In particlar, biologists often employ models comprised of polynomial ordinary differential eqations that arise from biochemical networs. Within this setting, the discrimination problem is cast as a finite-horizon, dynamic, zero-sm game in which parameter ncertainties in the model oppose the effort of the experimental conditions. The reslting problem, inclding some of its nown relaxations, is intractable in general. Here, a new scalable relaxation method that yields sfficient conditions for discrimination is developed. If the conditions are met, the method also comptes the associated random experiment that can discriminate between competing models with high probability, regardless of the actal system behavior. The method is illstrated on a biochemical networ with an nnown strctre. I. INTRODUCTION Systems biologists are often faced with competing dynamical models of experimental systems. For example, a newly discovered reglatory protein identified as an inhibitor may act on one of several possible genes in a pathway, leading to different possible biochemical networ models. One way to distingish between different models is to create nocot cell-lines and examine the steady-state behavior of the reslting mtants. However, sch experiments are tedios and may be qite difficlt depending on how vital the considered networ is to the normal operation of the cell. An alternative approach is to rn dynamical experiments on cells and interpret their transient response to distingish between competing models. For example, a ntrient or a chemical signal can be changed in a time-varying manner i.e. it is an inpt signal to the system) and the intensity of a florescent marer incorporated into the networ can be observed, as was recently demonstrated with osmotic pressre reglation in yeast [1]. Dynamical experiments are still expensive to set p, bt they are potentially less invasive and mch more informative than static experiments. Within this setting, we address the qestion: What experiments shold be performed on the physical system to ensre that as many candidate models are invalidated by the experimental reslts as possible? In particlar: given a set of candidate models, can we define a probability distribtion, which we call a disparity certificate, This wor was spported in part by the following grants: NIH/NIDCR, #T32 DE and ARO MURI, #W 911 NF D. Georgiev and E. Klavins are with the Department of Electrical Engineering, University of Washington, Seattle, WA , USA dgeorgie@.washington.ed, lavins@.washington.ed over the possible inpt signals that maximally distingishes the candidate models? If no inpt signals distingish the candidate models, the experiments are not worth doing ntil better candidates are derived. If the candidate models are distingishable, probing the actal experimental system with the reslting disparity certificate mst invalidate at least one of the models see Figre I). Invalidation is made difficlt by ncertainty in the experimental system. In systems biology, ncertainty arises particlarly from nnown and even time varying reaction rates e.g. some fnctions slow down dring cell division). Ths, an erroneos model may not be invalidated by an experiment becase the ncertain parts of the physical system conspire against the researcher to prodce otpts that seem consistent with the model. In or search for a disparity certificate we mst expect that ncertainty in the system will wor to mae candidate models indistingishable. The reslting problem becomes a game: the inpt signal tries to force the otpts of two candidate models to be different while the ncertainty tries to mae the otpts the same. Inpt M1 M2 Physical System Otpt A B C Time Fig. 1. Model Discrimination for experimental design. X represents the disparity certificate. Shaded regions represent all possible trajectories corresponding to the inpt. Trajectories A, B, C are possible otpts of the physical system. For otpt A model 2 is invalidated, for otpt B, model 1 is invalidated and for otpt C both models are invalidated. In this paper, we sppose we are given two different candidate models that have the same inpt and otpt spaces. We pt ncertainty in the initial conditions and parameters of the models. We pose two problems: Model Discrimination and Model Invalidation. The first problem we set p as a maximin problem maximizing over all inpt signal distribtions and minimizing over all distrbances the difference between the otpts of the two models). Since the reslting optimization problem is intractable, we introdce a relaxation based on the theory of moments [2]) that we can solve more efficiently. The second problem is then easily solved sing the disparity certificate prodced by the first problem. We have implemented the method in MATLAB and demonstrate

2 it on two problems involving candidate parameter regimes in biochemical networs. II. RELATED WORK Model discrimination is widely stdied in the literatre. Experimental design based on model discrimination has also received a considerable attention [3 7]. The traditional approach is to apply Bayesian conditioning and discriminate models sing criterion sch as the maximm lielihood [3, 8, 9]. Methods based on the Bayesian approach assme that the real world system behaves according to a given candidate model with a nown probability. As a reslt, the sccess of these methods depends on the qality of prior information. More recent methods are based on deterministic models [4 6], as is the case in this paper. In [6], a method is developed that comptes the initial state to maximize an pper bond on the distance between otpts of competing models. The models, however, are deterministic and, by maximizing the pper bond, the method cannot garantee discrimination. In [4] algorithms for efficiently discriminating models from experimental data are developed, however, no inpts are considered. Finally, a great deal of literatre on axiliary inpt design for falt detection deals with model discrimination [8, 10]. The majority of this wor is based on linear models and Bayesian conditioning. In or wor, nonlinear polynomial) systems of difference eqations with nstrctred ncertainty are considered and inpt signal distribtions garanteeing discrimination are compted. A. Informal Description III. PROBLEM STATEMENT The wor herein is motivated by biochemical experimental design. The following is an informal description of or wor; the formal description is in the next section. We consider models comprising polynomial difference eqations and linear otpts. An experiments controlled variables are modeled as control inpts to the system. We assme feedbac is not possible or practical dring the exection of the experiment and, therefore, restrict the control inpts to be fnctions of only time. An experiments ncontrolled variables are modeled as distrbances. Distrbances are nstrctred, i.e., can be effected by past events in an arbitrary way, and the only nown information abot them is their domains. Note that distrbance information is part of the model. The methods herein do not reqire that the actal experiments ncontrolled variables have the same domain as the distrbances. Comptation in experiments is performed in two steps: 1) experimental design and 2) data analysis. The two steps are matched by the two problems considered in this paper. The first problem considers feasibility of model discrimination and the comptation of discriminating inpts, called disparity certificates. Model Discrimination Problem MDP) informal). Given a pair of candidate models with the same inpt and otpt spaces, find an inpt, called the disparity certificate, that yields different otpts for all possible distrbances. If a disparity certificate exists, we can hope to implement it in the experiment and learn which model does not represent the system. Identifying the inconsistent model is called model invalidation. The second problem sees to invalidate the candidate models based on experimental data from a series of experiments. Model Invalidation Problem MIP) informal). Given the inpts and otpts for a series of exected experiments, find which candidate model maps the inpts to different otpts for all possible distrbances. Model invalidation is stdied in [11]. As will be shown, disparity certificates enable a model to be invalidated by simltaneosly comparing it with experimental data from mltiple trials. The method in [11] does not consider sch comparisons and the extension is not clear. We address the problem within the framewor of this paper and show that if we can find a disparity certificate, we can invalidate a model. B. Formal Description The following notation is sed. Lowercase letters are sed to denote fnctions and vectors. A seqence of vectors {r 1),..., r K)} is written as r K. Uppercase letters are sed to denote matrices and random variables. As with a seqence of vectors, a seqence of random variables {R 1),..., R K)} is written as R K. For any discrete random variable R, P r) is the probability that R = r the random variable associated with R will be clear from the context). For any seqence of random variables R K, P r K) r K 1) is the probability that R K) = r K) given that R K 1 = r. Uppercase script letters denote sets. For a given set R, R K is the cartesian prodct K i=1r. The set of all probability distribtions over R is denoted by R). For a vector β N K, we se the notation from [2] and write the monomial r β1 1 rβk K as rβ. Throghot this paper, we consider a pair of models described by discrete time difference eqations of the form { xi + 1) = f M i = i x i ), w i ), )), y i ) = C i x i ), where {1..., K} and i {1, 2}. The initial state satisfies x i 1) X i 1) R ni, the control inpt satisfies ) U R m, and the distrbance satisfies w i ) W i R di. Throghot the paper, we concatenate a pair of models to form a composite model M with state x = x 1, x 2 ) R n, state transition fnction f = f 1, f 2 ), control inpt, distrbance w = w 1, w 2 ) R d, and otpt y = y 1 y 2. We mae the following assmptions. Assmption A1. State transition fnction f is a polynomial fnction with respect x and w, i.e., f x, w, ) = α A2. The set of control inpts U is finite. 1) a α ) x, w) α. 2)

3 A3. The sets X 1) and W are compact intervals in their respective Eclidean spaces. Before we proceed to address the isse of model discrimination, we mst define how the vales of the control inpts and the distrbances are selected. Experimental data is often evalated after the completion of an experiment. Hence, we reqire that open loop control inpt be selected. Open loop evalation, however, does not imply that the inpt is selected deterministically. Indeed, random control inpts are commonly sed in system identification to provide persistent excitation [12], and in game theory it is well nown that random decisions often otperform deterministic decisions [13]. In this wor we consider random control inpts, and, throgh example, show that this increases the class of systems that can be discriminated. De to physical, experimental, or comptational constraints, not all possible trajectories in U K may be allowed. Define the set of control inpts allowed to follow by S ) U. The set of allowable control inpt trajectories of length is denoted by T and is defined recrsively by T i = { i U i i 1 T i 1, i) S i i 1 )}. Therefore, a control inpt is a random seqence U K with the sample space T K. Whether an inpt discriminates two models depends on how the distrbance is realized. The distrbance may be nnown bt fixed a priori. This wold imply that no matter what the inpt is, the ncertainty remains nchanged. If the inpts do affect the distrbance, the effect may be casal. This wold imply that the ncertainty can change if any of the past inpts are changed. If the inpts affect the ncertain parts in a noncasal manner, the ncertainty can change if any inpts past or ftre) are changed. Which of the three alternatives is chosen depends on the system. However, in biological systems, the ncertainty e.g., the reaction rates) is assmed to depend on the crrent state of the system and is therefore casal. Moreover, viable experiments are dplicable. Therefore, we assme that the distrbance at any time instance is a fnction of all past exogenos inpts to the system and this fnction is the same for different experimental trials. Note that the initial state is treated as part of the distrbance, which leaves the control inpts as the only exogenos inpts. The vale of the distrbance given that the control inpts p to time 1 are given by is denoted by w ; ). Similarly, x ; ) and y ; ) denote the state and otpt, respectively, that reslt from the control inpts and the distrbances w i; i 1), i {1,..., }. A random control inpt U yields the random distrbance w ; U ), the random state x ; U ), and the random otpt y ; U ). The disparity of a pair of models is defined in terms of a disparity certificate. Definition 1 Disparity Certificate). A pair of models M 1, M 2 ) is said to have a disparity certificate U K if, for all conditional distrbances w ; ) W, {1,..., K}, the otpt is nonzero with positive probability, i.e., for some {1,..., K}, P y ; U ) 2 > 0 ) > 0. The two problems considered in this paper can now be formally defined. The model discrimination problem is cast as a dynamic game where the distrbance ses information abot all past control inpts to try and eep the otpts of the two models eqal. Model Discrimination Problem MDP). Given a composite model M, find a U with the distribtion P that solves the following maximin problem: max P sbject to min w K T K P K ) K+1 =1 y ; ) 2, x + 1; ) = f x ; ), w ; ), ) ) y ; ) = Cx ; ). Note that the cost in MDP is jst the expected vale with respect to the control inpt U. A disparity certificate exists if and only if the optimal cost in MDP is positive. Note that the polynomial constraints mae MDP nonconvex in general. Next we address the model invalidation problem. Sppose that l trials of a particlar experiment have been performed. We wold lie to now if one of the models is inconsistent, and thereby invalidated, by the trials. Recall that the conditional distrbances are assmed to be the same in different trials. Therefore, if a model is consistent, it shold be able to approximate otpts from all trials at the same time. Model Invalidation Problem MIP). Let the reslts of l experimental { trials be given by the set of inpt-otpt pairs )} K e,1, ye,1) K,..., K e,l, yk e,l. For a given model M i, the model invalidation problem is defined as follows. l K+1 minimize ) yi ; w e,j ye,j ) 2, 2 j=1 =1 sbject to ) ) ) x i + 1; e,j = fi xi ; e,j, wi ; e,j, e,j ) ) ) ) y i ; e,j = Ci x i ; e,j. If the optimal cost of MIP is positive for model M i, then the model is invalidated. Notice that MIP also presents a nonconvex optimization problem. The following reslt follows directly from Definition 1, MDP, and MIP. Proposition III.1. Let a pair of models M 1, M 2 ) have a disparity certificate U. Let SUP U ) be the cardinality of the sample space of U. Then at most SUP U ) different experiments are reqired to invalidate at least one model. Therefore, the existence of a disparity certificate implies that, given enogh experimental trials, sefl information i.e., invalidation of a model) mst eventally reslt. Note that experiments are typically expensive and difficlt to

4 organize bt execting mltiple parallel trials is relatively easy. IV. SOLUTION To discriminate two models, one mst solve MDP. First, for any control inpt U, the corresponding global minimizer the worst distrbance) w mst be fond. However, finding a global minimizer in MDP is in general nmerically intractable de to the polynomial state transition eqality constraints. Second, the maximizing control inpt mst be fond. The first part of this section develops a method for compting the lower bond on the optimal cost for a given control inpt. The second part of the section appends this method to allow for simltaneos comptation of the lower bond maximizing control inpt. A. Worst Distrbance Several methods exist for minimizing the cost in MDP. The brte force method is to discretize the state space and nmerically solve the Bellman eqation, see [14] for examples. The state space dimension of the class of systems considered herein prohibits sch an approach. Other methods yield a lower bond on the cost. One sch approach is to se difference inclsions to compte the set of reachable states and then minimize the cost over all trajectories [15]. Difference inclsions however are overly conservative in that ftre states depend only on the crrent set of reachable states and not on the actal crrent state vale. Another approach is to approximate the global optimal soltion sing smof-sqares convex relaxation [2, 16]. This approach yields a lower bond on the optimal soltion p to any specified tolerance. However, it is only nmerically feasible for problems comprising relatively few optimization variables; the method presented in [2] applied to MDP with 10 time steps, and 5 dimensional state space and 5 dimensional distrbance space at each time step reqires approximately 100 million optimization variables for a first order relaxation. In this section, we constrct a method for compting the lower bond on the optimal cost sing the theory of moments. The method yields a convex relaxation that is nmerically feasible and retains some state transition information. It is not garanteed to prodce the actal lower bond, it does allow for better approximations throgh higher order relaxations. The reslting convex problem is solvable sing standard interior point methods. The method developed herein is based on the following lemma from [2]. Lemma 1. For a given cost fnction J : R n R, the following two optimization problems are eqivalent: minimize J x) minimize J x) dµ x). x X µ X) To illstrate how Lemma 1 is tilized, consider the following nonlinear, scalar difference eqation for either candidate model: x + 1) = x ) + x ) 2 w ). 3) X Let the real state and distrbance variables x) and w) be the random variables X) and W ) which is withot loss of generality in MDP and MIP by Lemma 1). Eqation 3 can then be looed at as a moment relation, and, instead of propagating state, we can propagate state moments. This relaxes the constraint imposed by the difference eqation and reslts in a lower bond on the cost in MDP. Higher moments can be compted to improve the bond bt at the expense of comptational costs. To see how we can propagate the first moments of the state, let the reachable set of x) lie in the set X ) = [x, x + ] and let the domain of w) be given by the set W = [w, w + ]. Then tae the expected vale of both sides of the random difference eqation to yield E [X + 1)] = E [X)] + E [ W )X) 2]. 4) At time, the vale of E [X)] is nown since it was compted at time 1. The vale of E [ W )X) 2] is not nown, since we don t now the fll distribtion of X). We can, however, constrain the vale sing the theory of moments) since we now the reachable set X) and the distrbance domain W. The constraints on the mixed moment are given in [2] and briefly reviewed here in the following lemma. Some additional notation is reqired. For a random vector X and a positive integer N, let X 1 N be the vector of all monomials p to the order N, e.g., X 1, X 2 ) 1 2 ) = 1, X1, X 2, X1 2, X 1 X 2, X2 2, and let X N N = X N 1 X 1 N. Frthermore, for an interval [x, x + ] that contains the spport of X, define the bloc diagonal matrix as X± N N = diag X i x ) ) i x + i X ) i X N N. Lemma 2. The expected vales of X N N and X N N ± are positive semidefinite. Lemma 2 is sed to constrain the mixed moment in Eqation 4 as follows. Denote the first moment of E [X] by x. Expand the dimension of the distrbance space by assigning a new distrbance variable w α1,α 2 to every monomial E [X α1 W α2 ]. Note that the distrbances corresponding to the first moment of X are not free, i.e., w 1,0 = x. Finally form the matrices in Lemma 2 ot of the distrbances w and the state x. What reslts is a convex approximation of Eqation 3 with the form where x + 1) = w 2,1 ) + x) moment diff. eq.), x + 1) X + 1) reachable stte), N ) 0 moment matrix), ) 0 constraint matrix), N 1,± N [X), ] ) := E W )) N N, [ ] N 1,± ) := E X), W )) N N. To see how these constraints affect the transition of x, see Figre 2. The figre plots a sample trajectory generated ± 5)

5 from random distrbance vales. At each point along the trajectory, we compted the set of possible vales for the next state. These vales are shown for the time = 7. The moment constrained dynamics close approximate the actal dynamics, while reachable set compte by solving the difference inclsion greatly overapproximates the actal dynamics. x) Actal Constraint Moment Constraint Reachable Constraint Sample Trajectory Potential Transitions Fig. 2. A plot of a random trajectory for the simple model in Eqation 3. At each point along the trajectory a set of possible ftre states can be compted sing the moment constraint, the reachable set constraint, and the actal difference eqation constraint. At time 7 the sets are shown in the figre. The approach sed for the system in Eqation 3 can be directly extended for the compose model M, except now separate moment and constraint matrices will be formed for each monomial. Moreover, the matrices will have to also be conditioned on the control inpt trajectory. Recall that the transition fnction for a fixed control inpt has the form f x, w) = α a α) x, w) α. Allow the state and the distrbance to be random variables X and W, respectively, and denote the mixed moment E [X, W ) α ] by w α. The domains of X and W are assmed to be some nown intervals. For i {1,..., n}, let w i = x i, and let A = a i )) i {1,...,n}. Then, order the remaining distrbances those not corresponding to first moments of X) into a vector w and let B be the matrix with colmns formed from the corresponding coefficients a α ). Then the difference eqation conditioned on can be approximated by the by the following convex constraints denoted by C ; ) ) x + 1; ) = A ) x ; ) +B ) w ; C ), ; ) x + 1; ) X + 1; ), ȳ ; ) = C x ; ), V ) 6) N α ; 0, V ) N 1,± α ; 0. MDP with the exact difference eqation constraint replaced by the constraints C ; ) is convex with respect to the distrbance. Moreover, if we fix the control inpt, the minimization problem is a linear matrix ineqality problem, which can be solved efficiently. Finally, if we replace the exact difference eqation constraint with the constraints C ; ) and then solve MDP the reslting minimization problem will yield a lower bond on the optimal cost. Therefore, if the optimal cost for the relaxed problem is positive for some U, then their exists a disparity certificate for the pair of models. B. Disparity Certificate Derivation The minimization problem with the relaxed transition constraints C ; ) can be solved sing standard interior point methods. In this section, we address the maximization over the control inpts for the relaxed problem. It is important that we find the optimal control inpt since the relaxed minimization problem is already conservative. First we show how the minimization problem can be modified to simltaneosly accont for the maximization over the control inpts. Second we show how to iteratively compte the optimal control inpt. Let MDP be the problem obtained by sbstitting the constraint C ) for the difference eqation constraint in MDP. This problem is convex in the distrbance variables when the probabilities are held fixed. Moreover, the cost fnction is concave linear) with respect to the probabilities P K). Since the distrbances and probabilities belong to convex, compact sets, it is nown that the problem has a saddle point, i.e., the max and min operations can be reversed [17]. Consider the following problem. Convex Model Discrimination Problem CoMDP). Given a composite model M, the problem is minimize λ1) + ȳ 1) 2 2, w,λ sbject to λ ; ) λ + 1; ) ȳ + 1; ) C ; ), for all {1,..., K}, K T K. CoMDP is eqivalent to the minimax problem where the control inpts can depend on the past and present vales of the states and distrbances. The existence of a saddle point for MDP implies the following reslt. Proposition IV.1. The optimal cost of MDP is eqal to the optimal cost of CoMDP. CoMDP can be solved sing standard interior point methods by forming a log barrier fnction for each ineqality constraint. The reslting cost fnction at each centering step then has the form J w) = K+1 l =1 T x ; ), w ; )), and is sbject to the eqality constraint C ). Efficient stepwise Newton s method algorithms can then be sed to minimize J. Sch algorithms have the favorable property of linear complexity growth with respect to the nmber of time steps. Next, we compte a control inpt U sfficient for discrimination. Let P ) be the conditional probability that U =, given that U =. Let w be the optimal distrbance vale. Define the partial cost evalated along the trajectory reslting from w as v ) = ȳ ; ) max S ) v +1, ).

6 Define the set of maximizing control inpts as Z ) { = argmax v, )}. Then, a control inpt that will discriminate the pair of models is given in the next theorem. Theorem 1. Sppose the existence of a disparity certificate is shown by solving CoMDP, then the following control inpt will discriminate the pair of models with positive probability: P ) { 1 = Z ), Z ). 0, otherwise Proof: All we need to show is that the optimal control inpt mst have the same spport as U K given by the transition probabilities P ) described in the theorem. Sppose that for some and, the optimal control inpt assigns a positive probability to a not in Z ). Then the expected cost will be less than v 1, which contradicts Proposition IV.1 that says the optimal cost is eqal to v 1. It is important to note that althogh Theorem 1 yields a control inpt that will discriminate the models, it is not the optimal control inpt. To compte the maximizing control inpt in MDP, frther development is reqired. Remar 1. If CoMDP prodces a disparity certificate and different control inpts from the sample space of U in Theorem 1 are applied to the experimental system in mltiple trials, then at least one model mst be invalidated in a fine nmber of trial by solving MIP MIP with the convex constraints C ; ) ). V. EXAMPLES In this section we introdce a two examples that illstrate the methods developed herein. A interior point method based on log barrier fnctions was developed that implements the stepwise Newton s method [18] to solve the centering problems. No special precations were taen to deal with ill posed Hessian matrices, hence preventing large problems from being solved. It is important to note, however, that this is not a limitation of the method, rather a limitation of the crrent nmerical implementation. A. Example: Inhibition Figre 3a) shows a biochemical reaction networ in with two species A and B. The inpt signal is the rate at which B is added and the otpt is the concentration of A. Two models of this networ are presented in the table of Figre 3a), the only difference being the range for the rate 2. The chemical inetics model corresponding to the networ is defined by [A] = 1 2 [A] [B] 3 [A], [B] = 2 [A] [B]. We model the ncertainty in the reaction rates as a distrbance as in Eqation 1, which implies that the reaction rates are allowed to vary with time. For the inpt, we consider 7) implses at specific time instances. To apply or method, we se the discrete time approximation of Eqation 7 is [A] + 1) = [A] ) + δ 1 ) ) 2 ) [A] ) [B] ) + ) 3 ) [A] ), 8) [B] + 1) = [B] ) + ) ) + δ 2 ) [A] ) [B] ) + ). The maximm inpt is = 1 and the minimm inpt is =.1. Or method shows that the two models can be discriminated. The reslting disparity certificate is shown in Figre 3b). For this system, a niform distribte over inpt signals is enogh to distingish between the compete models. In Figre 3c) we illstrate the otpt of model one for a random pair of inpts and show that in the worst case, model two cannot match the otpts and hence is invalidated with respect to model one). B. Example: Self-Annihilation Figre 4a) shows a networ in which a species A activates its own degradation. The two candidate models for the system differ in the range they assign to the parameter 2, and ths to the degree to which A annihilates itself. Figres 4b) and c) are similar to those of the previos example, except that here the disparity certificate weights some inpt signals more than others, therefore sggesting a different array of experiments with the physical system. VI. CONCLUSION Model discrimination and model invalidation for polynomial system was considered in the framewor of dynamic games. The class of disparate competing models was broadened throgh the se of stochastic inpts, i.e., models that no one inpt can discriminate, were shown to have a disparity certificate. The model invalidation problem for this setp was formlated. It was shown that at least one of two disparate models can be invalidated in a finite nmber of experiments. To solve the model invalidation and discrimination problems, a scalable convex relaxation method based on the theory of moments) was developed and implemented in MATLAB. Ftre wor incldes a more efficient implementation of the developed theory and experimental application. REFERENCES [1] J. T. Mettetal, D. Mzzey, C. Gmez-Uribe, and A. van Odenaarden, The freqency dependence of osmo-adaptation in saccharomyces cerevisiae, Science, vol. 319, no. 5862, [2] J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM Jornal on Optim., vol. 11, no. 3, [3] R. Horn, Statistical methods for model discrimination. Applications to gating inetics and permeation of the acetylcholine receptor channel, Biophysical Jornal, vol. 51, no. 2, [4] R. Feely, M. Frenlach, M. Onsm, T. Rssi, A. Arin, and A. Pacard, Model discrimination sing data collaboration, Jornal of Physical Chemistry A, vol. 110, [5] J. F. Apgar, J. E. Toettcher, D. Endy, F. M. White, and B. Tidor, Stimls design for model selection and validation in cell signaling, PLoS Comp. Biol., vol. 4, no. 2, 2008.

7 M 1 M B A 3 A1) B1) [1.0, 1.1] [0.3, 0.4] [0.8, 1.2] [0.0, 0.5] [0.5, 1.0] [1.0, 1.1] [0.3, 0.4] [0.8, 1.2] [.75, 1.0] [0.5, 1.0] P M ) = y 1 -y 2 = Pm)= Σy 2 y), y e ) Disparity Certificate y e 1 ): otpt of experiment 1 y e 2 ): otpt of experiment 2 y 1 1 ) joint otpt of model 1 y 1 2 ) closest to y e 1 ), y e 2 )) a) b) c) Fig. 3. a) A simple biochemical networ in which a the prodction of a species A is inhibited by a species B. The inpt to the system is the rate at which B is added to the system. The two models for this system have identical strctres, bt different ranges for the rate 2 b) The compted disparity certificate. The tree traces all inpt trajectories. Each branch corresponds to a control inpt selection. Upper branches correspond to maximm inpt and lower branches correspond to minimm inpt. The transition probabilities corresponding to the disparity certificate are shown at their respective branches. For each time instance, the absolte vale of the scaled factor of 100) difference between y 1 and y 2 is specified. Following the leaves of the tree are shown the smmed sqare costs. c) Sample trajectories are of model one and the closest model two trajectory are shown. The plot indicates that if the physical system behaves according to model one, then model two will always be invalidated given enogh experimental data. 1.7 Σy 2 =3.89 M 1 M 2 A 2 1 A1) 1 2 [.95, 1.05] [.75, 1.0] [0.5,.75] [.95, 1.05] [.75, 1.0] [0.0,.25] P M ) =0.28 = M y 1 -y 2 =.61 =m.22 Pm)= =3.89 =4.12 =4.12 =4.16 =4.16 =4.20 =4.20 y), y e ) Disparity Certificate y e 1 ): otpt of experiment 1 y 1 1 ): otpt of model 1 closest to y e 1 ) a) b) c) Fig. 4. a) A biochemical networ in which a the degradation of a species A is activated by A itself. The two models for this system have identical strctres, bt different ranges for the rate 2 b) The compted disparity certificate. c) Model one sample trajectories. The closest model two trajectory to the model one trajectory is also shown. The plot indicates that if the physical system behaves according to model one, then model two will always be invalidated given enogh experimental data. [6] A. Papachristodolo and H. El-Samad, Algorithms for discriminating between biochemical reaction networ models: Towards systematic experimental design, in Proc. of the American Control Conf., [7] A. Kremling, S. Fischer, K. Gadar, F. J. Doyle, T. Sater, E. Bllinger, F. Allgower, and E. D. Gilles, A benchmar for methods in reverse engineering and model discrimination: Problem formlation and soltions, Genome Research, vol. 14, [8] L. Blacmore and B. Williams, Finite horizon control design for optimal model discrimination, in Proc. IEEE Conf. Decision & Control, [9] K. Felsenstein, Optimal Bayesian design for discrimination among rival models, Comp. Stat. and Data Anal., vol. 14, no. 4, [10] T. Hatanaa and K. Uosai, Optimal axiliary inpt for falt detection of systems with model ncertainty, in Int. Conf. on Contr. App., [11] S. Prajna, Barrier certificates for nonlinear model validation, Atomatica, vol. 42, no. 1, [12] R. D. Nowa and B. D. V. Veen, Random and psedorandom excitation seqences for nonlinear system identification, in IEEE Winter Worshop on Nonlinear Digital Signal Proc., [13] R. B. Myerson, Game Theory, Analysis of Conflict. Harvard University Press, [14] I. M. Mitchell, Application of level set methods to control and reachability problems in continos and hybrid systems. PhD thesis, Stanford University, Dept. of Scientific Compting and Comptational Mathematics, [15] T. A. Henzinger, B. Horowitz, R. Majmdar, and W. Howard, Lectre Notes in Compter Science, ch. Beyond HYTECH: Hybrid Systems Analysis Using Interval Nmerical Methods, pp Springer- Verlag, [16] P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming, vol. 96, no. 2, [17] T. Basar and G. Olsder, Dynamic noncooperative game theory. SIAM Classics in Applied Mathematics, second ed., [18] J. C. Dnn and D. P. Bertseas, Efficient dynamic programming implementations of Newton s method for nconstrained optimal control problems, Jornal of Opt. Theory and App., vol. 63, no. 1, 1989.