Monte Carlo Observer for a Stochastic Model of Bioreactors


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1 Mone Carlo Observer for a Sochasic Model of Bioreacors Marc Joannides, Irène Larramendy Valverde, and Vivien Rossi 2 Insiu de Mahémaiques e Modélisaion de Monpellier (I3M UMR 549 CNRS Place Eugène Baaillon 3495 Monpellier cedex 5 France ( 2 Laboraoire de Biosaisique Insiu Universiaire de Recherche Clinique 64 avenue du Doyen Gason Giraud 3493 Monpellier cedex 5 France ( Absrac. This paper proposes a (sochasic Langevinype formulaion o modelize he coninuous ime evoluion of he sae of a biological reacor. We adap he classical echnique of asympoic observer commonly used in he deerminisic case, o design a Mone Carlo procedure for he esimaion of an unobserved reacan. We illusrae he relevance of his approach by numerical simulaions. Keywords: Biochemical Processes, Sochasic Differenial Equaions, Mone Carlo, Observer. Inroducion We are ineresed in monioring he sae of a biological reacor, which is basically a ank in which microscopic living organisms consume a nurien. Monioring he process is ofen a criical issue in many indusrial applicaions, as i is a firs necessary sep owards is conrol. However, only a few of he componens of he sae (he insananeous composiion of he reacor are measured by sensor. In paricular, he concenraions of he biomass are generally no available on line, ye hey are of he greaes imporance for he process conrol. This problem led o he design of observer as sofware sensor, which are esimaors of he unobserved componens based on he available measuremens. The reconsruced sae is hen used in he conrol process as if i were compleely observed. A review of he commonly used echniques can be found in [Basin and Dochain, 99]. Observers found in he lieraure can be divided in hree disinc classes: he observers obained from he general heory (Kalman like and nonlinear observers exploi all he knowledge given by he model, including
2 2 Joannides e al. he kineics par. However, modeling he biological kineics reacion is a difficul ask, so ha he model used by he observers could differ significanly from he realiy. This resuls in a (possibly imporan esimaion bias. The asympoic observers make use of a specific feaure of he bioprocesses model, relaed o he noion of reacion invarian. The idea is o design an observer for he oal mass of he componens involved in he biological process, and hen o reconsruc he whole sae wih his observer and he measured componens. This approach circumven he knowledge of he kineics bu is rae of convergence highly depends on he operaing condiions. Observers ha lie somehow in beween hese wo classes are based on a parial knowledge of he kineics. They use a parameric model of he reacion kineics and aemp o esimae he parameers ogeher wih he sae iself. All of hese approaches have been implemened for various indusrial applicaions ([Dochain, 23]. I should be noed ha, exceped for he Kalman filer, hese echniques were mainly developed in a compleely deerminisic conex. Uncerainies in he modelling are accouned for only hrough varying parameers, and performance in he presence of noisy inpus or measuremens are evaluaed by numerical simulaion. The aemps o ackle hese problems are very few. We should menion he inerval observer of [Rapapor and Dochain, 25] which uses he noion of cooperaiviy o produce bounds for he asympoic observer, when he dynamics and he inpu are uncerain. On he sochasic side, [Rossi and Vila, 25] proposed a formulaion as a filering problem. Sochasic erms are inroduced in he dynamics, he measuremens and he iniial condiion. The objec of ineres is hen he condiional probabiliy law of he whole sae given noisy measuremen of some componens. The model considered was obained by adding a discree ime whie noise o he deerminisic model. The approach presened in he presen paper consiss in modeling he uncerainies on he dynamics by a sochasic differenial sysem in a way ha is consisen wih he noion of invarian. We hen design a se of asympoic observers which is used, ogeher wih he observed componens o approximae he probabiliy law of he unobserved ones. Sae esimaion, variances, bounds and confidence regions are obained from his Mone Carlo approximaion. Recall firs he classical model obain from he mass balance principle, for a coninuous sirred ank reacor : ( (ḃ = r(b c, s D ṡ ( b s + D ( s in (
3 Mone Carlo esimaion for biochemical processes 3 where he subsrae S is consumed by a biomass B wih yield coefficien c, he biological reacion being represened by S c B. Here b and s denoes he concenraion of biomass B and subsrae S respecively, D is he diluion rae, r(. he reacion kineics and s in he subsrae concenraion in he inle. Many forms for he reacion rae have been proposed in he lieraure. The mos commonly used is he Monod model s r(b, s = µ max k S + s b. 2 Sochasic model The deerminisic dynamics ( is he sum of hree vecor fields represening he acion of hree sources of variaions (he biological reacion iself, he flow ou and he inle, each of which being subjec o random disurbances. Therefore, if noise erms are o be inroduced, hey should affec independenly he hree differen direcions. The random naure of biochemical reacion a he molecular scale has been menioned and sudied by many auhor, see [Gillespie, 2] or [El Samad e al., 25]. A a macroscopic scale, [Kurz, 978] modelled he overall effec of hese individual reacions on he global concenraions, by an addiive noise erm of variance proporional o he reacion kineics (or propensiy funcion r. In his conex, he sae (B, S is hen a Markov process saisfying he Langevin chemical equaion d ( B, S = ( c ( r(b r(b, S d +, s c dv where V denoes a Wiener Process. Noice ha he drif and he diffusion coefficiens ac in he same direcion. On anoher side, he flow in and ou may also be more precisely described by a sochasic dynamics. Indeed, he diluion rae D denoes acually he average value, ignoring he inhomogeneiy of he medium. Adding (formally independen whie noises o D in he flow in and ou, we ge a sochasic dynamics for he exchange wih environmen: (D + ( W ou B S + (D + W in ( s in Puing all his ogeher, we ge our sochasic model d ( [( ( ( ] B B, S = r(b c, S D + D S s in d ( ( ( r(b B +, s c dv + dw ou S + s in. dw in (2
4 4 Joannides e al. We observe ha nonlineariy in model (2 lies only in he reacion kineics. We herefore define he oal mass proccess Z = B + c S and noe ha his quaniy remains unchanged hrough he biological reacion. The dynamics of Z, obained from (2 is hen linear dz = D (Z Z in d + Z dw ou + Z in dw in. (3 I is worh noing ha his linear SDE has an explici soluion ([Klebaner, 998] where Z = Z + D U zs in U ds + U s zs in U s dw in s U = exp{ (D + σ2 ou 2 + W ou }. Finally, noice ha he linear change of coordinae (b, s (b + s c, s, gives he equivalen model (Z, S, where Z has a linear dynamics, which is independen of he reacion kineics r(. This feaure can be exploied o design efficien simulaion algorihms. In he nex secion, we will use i o produce an esimaion of he unknown concenraion B, based on he compleely observed concenraion S. 3 Mone Carlo approximaion A possible adapaion of he asympoic observer approach o his sochasic model could be done as follows: Generae iniial condiions for a se of N independen asympoic observers hereafer named paricles. Le each paricle evolve independenly according o dynamics (3, up o ime. Deduce a se of observers B using he observed componen S. Le Q denoes he law of he mass balance process Z. Iniializaion Firs of all, o approximae iniial condiion Q, we begin simulaing an approximaion of he iniial condiion P, he law of (B, S. Le {(b i, s i, i =,.., N} be an Nsample disribued according o P. We define he Mone Carlo approximaion of P by P N = N δ (b i,s i. I follows from he law of large numbers ha if N is large enough hen P N will be a good approximaion of P in he sense ha for each φ bounded measurable P N, φ P, φ N
5 Mone Carlo esimaion for biochemical processes 5 We hen deduce he Mone Carlo approximaion of Q by Q N = N δ z i where z i = b i + si c. Evoluion We hen generae N independens soluions of (3, saring from he N iniial condiions {z i }. Denoing by {z i, i =,..N} his se of soluions, we define he empirical measure: Q N = N δ z i. Back o he original model Finally, we ranslae his empirical law Q N, making use of he observaion S. Le {b i = z i S c, i =,.., N} and define µ N = N δ b i. The unobserved componen B is esimaed by he average B = N b i. Noice ha µ N provides us wih oher useful saisics like variance or mode. 4 Numerical resuls We firs illusrae he behaviour of he sochasic model hrough numerical simulaion, using a Monod model for he growh rae and a consan influen concenraion s in. The following able shows he values of he parameers: c µ max k S D s in 2.33 h 5 g/l.5 h 5 g/l Table. Parameer values We use he Euler Maruyama scheme o simulae he soluions of all he SDE involved, see ([Kloeden and Plaen, 992]. Fig. shows a ypical rajecory of he sochasic sysem saring from he equilibrium of he corresponding deerminisic model (2.54,.893. As
6 6 Joannides e al. (B,S S B Z Fig.. Sysem evoluion saring from equilibrium expeced, he diffusion coefficien is predominan so ha he sysem keeps oscillaing in a neighbourhood of he equilibrium sae. Fig. 2 shows a rajecory for he same sysem iniialized wih a value far from he equilibrium. We observe he predominance of he drif coefficien, which draws he sysem near he equilibrium, which is he righ behaviour. Reasonable changes in parameers does no affec his global picure. Performance of he Mone Carlo asympoic observer is illusraed by Fig. 3 and Fig. 4, which use observaion s from Fig. and Fig. 2 respecively. We represen he paricle cloud {b i }N ogeher wih is densiy esimaion. The rue sae is spoed by he verical line. Observe ha, since he paricle cloud moves according o he general dynamics, i follows closely he rue value. Indeed, each paricle is an individual asympoic observer, i.e. a poenial sae of he unobserved componen. Therefore, he esimaed densiy is, in some sense, a summary of our knowledge on his componen based on observed par. 5 Conclusion We have esablished he relevance of a coninuous ime sochasic modelizaion in bioechnology, ha is able o ake advanage of he exising know how in biology and opimizaion. We have successfully shown he feasibiliy of his approach by numerical simulaions. A leas wo direcions for fuure
7 Mone Carlo esimaion for biochemical processes 7 (B,S S B Z Fig. 2. Sysem evoluion saring far from equilibrium esimaed densiy B Time: esimaed densiy B Time: Fig. 3. Mone Carlo observer  equilibrium esimaed densiy B Time: esimaed densiy B Time: Fig. 4. Mone Carlo observer
8 8 Joannides e al. invesigaions are generalizaion o higher dimensional models (p reacions involving q reacans and a more realisic reamen of observaions. Indeed, we have supposed here ha one reacan was observed wihou noise, in he manner of opimizers, whereas his is clearly no he case in pracical siuaions. Paricle filers would surely be more appropriae o he case of noisy observaions. References [Basin and Dochain, 99]G. Basin and D. Dochain. Online esimaion and adapive conrol of bioreacors. Elsevier, Amserdam, 99. [Dochain, 23]D. Dochain. Sae and parameer esimaion in chemical and biochemical processes: a uorial. Journal of Process Conrol, 3(8:8 88, December 23. [El Samad e al., 25]Hana El Samad, Musafa Khammash, Linda Pezold, and Dan Gillespie. Sochasic modelling of gene regulaory neworks. In. J. Robus Nonlinear Conrol, 5(5:69 7, 25. [Gillespie, 2]Dan T. Gillespie. The chemical langevin equaion. J. Chem. Phys., 3:297 36, 2. [Klebaner, 998]Fima C. Klebaner. Inroducion o Sochasic Calculus wih Applicaions. Imperial College Press, London, Sepember 998. [Kloeden and Plaen, 992]P.E. Kloeden and E. Plaen. Numerical Soluion of Sochasic Differenial Equaions, volume 23 of Applicaions of Mahemaics. Springer Verlag, New York, 992. [Kurz, 978]Thomas G. Kurz. Srong approximaion heorems for densiy dependen Markov chains. Sochasic Processes Appl., 6:223 24, 978. [Rapapor and Dochain, 25]A. Rapapor and D. Dochain. Inerval observers for biochemical processes wih uncerain kineics and inpus. Mah. Biosci., 93(2: , January 25. [Rossi and Vila, 25]Vivien Rossi and JeanPierre Vila. Filrage de bioprocédé de dépolluion. approche par convoluion pariculaire. esta, 2(, 25.
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