journa l homepage: www.e lsev ier .com/ locate /b iosystems
odeling acclimatization by hybrid systems: Condition changeslter biological system behavior models
odrigo Assard,b,∗, Martín A. Montecinob,c, Alejandro Maasse,c, David J. Shermana
INRIA Bordeaux Sud-Ouest, Project-team (EPC) MAGNOME common to INRIA, CNRS, and U. Bordeaux 1, Talence, FranceCentro de Investigaciones Biomédicas, Facultad de Ciencias Biológicas and Facultad de Medicina, Universidad Andrés Bello, Santiago, ChileFONDAP 15090007 Center for Genome Regulation, Santiago, ChileICBM Instituto de Ciencias Biomédicas, Facultad de Medicina, Universidad de Chile, Santiago, ChileDepartamento de Ingeniería Matemática y Centro de Modelamiento Matemático, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile,antiago, Chile
r t i c l e i n f o
rticle history:eceived 26 July 2013eceived in revised form 6 March 2014ccepted 28 May 2014vailable online 2 June 2014
a b s t r a c t
In order to describe the dynamic behavior of a complex biological system, it is useful to combine modelsintegrating processes at different levels and with temporal dependencies. Such combinations are neces-sary for modeling acclimatization, a phenomenon where changes in environmental conditions can inducedrastic changes in the behavior of a biological system. In this article we formalize the use of hybrid systems
eywords:cclimatizationybrid systems
as a tool to model this kind of biological behavior. A modeling scheme called strong switches is proposed.It allows one to take into account both minor adjustments to the coefficients of a continuous model,and, more interestingly, large-scale changes to the structure of the model. We illustrate the proposedmethodology with two applications: acclimatization in wine fermentation kinetics, and acclimatizationof osteo-adipo differentiation system linking stimulus signals to bone mass.
Biological systems are known for their high degree of complex-ty. System behavior depends on many factors that are often notontrolled, and biological phenomena arise from different inter-cting processes. Different processes may be best described byifferent models: continuous models to describe gradual changesver time, discrete models for instantaneous changes, determinis-ic models to represent predictable behaviors, non-deterministic
odels to describe various possible responses, and stochastic mod-ls to introduce randomness (Wilkinson, 2006). Such is the case inhe modeling of a population of cells, in which one has to defineirths, growths, divisions and deaths of cells; at each cell one
escribes the transport relations between compartments such asytoplasm, mitochondrion and nucleus; and finally to model bio-hemical reactions responsible for metabolism (Maus et al., 2008).
∗ Corresponding author at: ICBM Instituto de Ciencias Biomédicas, Facultad deedicina, Universidad de Chile, Santiago, Chile. Tel.: +56 229786463;
In addition to this complexity, at each level the system behavior canconsiderably change according to environmental conditions, a phe-nomenon defined as acclimatization (Varela et al., 1974; Watts et al.,1975; Coles and Brown, 2003). Acclimatization behaviors can affectsystem dynamic laws as well as the model itself. In some cases, theeffect of such changes can be captured by model coefficients. How-ever, in some situations strong effects can be induced in the systembehavior, rendering coefficient changes insufficient. In fact, whendifferent conditions are described by different types of models, thedecision to unify models structurally different is expensive in termsof time and rewriting.
According to Minsky (1968), a model responds to questionsabout a system in a specific condition. To build a model that is validin general conditions and that takes acclimatization into account,it is necessary to reuse and combine models at different regimes.Unfortunately, unifying all the existing models is expensive interms of time and rewriting, and sometimes is not even meaningful.Furthermore, while efforts for finding a way to describe biologicalmodels that may be reused by third parties has finally established acommon format in SBML (Hucka et al., 2010), the needs for combin-
ing models and simulation have not yet been sufficiently covered.Recently, research on combination of models has been intense (seethe book sections Uhrmacher et al. (2005) and Maus et al. (2008)).Reusability and unambiguity have been identified as important
Fig. 1. Two study cases: Biological systems for wine fermentation kinetics andosteo-adipo differentiation. Systems acclimatize according to environmental con-ditions. When the resources are scarce yeasts compete and osteo-adipo precursor
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hallenges, but, to the best of our knowledge, there is no for-alization of the combination process. Here we approach these
hallenges. In this study, we propose a hybrid system-based protocolo model complex biological systems, emphasizing acclimatization.n this context, system dynamics is represented by the variation ofontinuous variables described by a model, whose laws can changeheir form over time depending on discrete mode changes. Theseast changes can be stochastic, non deterministic or deterministic.lthough it is well known that hybrid models are useful in biologynd medicine1 (Aihara and Suzuki, 2010), their relation to acclima-ization has not been explored in depth and modeling protocols areimited. In best of our knowledge, most of publications focus on par-icular applications, or in theoretical aspects of the formalizationnd model checking using examples as illustration.
Although hybrid automata appears as the accepted formalescription of hybrid systems, different implementations haveeen used depending on the particular application. Thus, someuthors utilize the COPASI package (Hoops et al., 2006) to simulateBML models of biochemical networks, but other researchers con-ider Matlab codes or they define new frameworks (see for exampleio-PEPA Ciocchetta and Hillston (2009) or CellExcite Bartocci et al.2008)). They neither allow reusing a priori defined SBML modelsithout rewriting them nor allow the possibility of different levels
f behavior changes.In order to consider systems with acclimatization, we adapt the
lassical idea of hybrid systems by allowing different levels of modehanges. We consider mode changes associated with environmen-al condition variations that modify the system behavior and whichhange the form of the continuous models. We propose two mod-ling schemes to describe the extended hybrid systems, namedoefficient switches and strong switches.
In a more theoretical perspective, our protocols also allow us toonsider the general problem of reconciliation of models for a givenrocess, where one needs to combine models and choose themynamically according to some external input. We show throughn example how reconciliation allows us to build models validor more general conditions than those described by the original
odels separately.These two schemes correspond to concrete cases encountered in
odel reuse. Acclimatization is often accompanied by physiolog-cally expensive and perhaps irreversible morphological changeso the individual, and a strong switch corresponds to this commit-
ent. Furthermore, in model reuse, it is important to recall thathe existing models have been experimentally validated at consid-rable expense. Modifying the existing models to combine themesults in a new model that must be revalidated. Using a coeffi-ient or a strong switch, however, preserves the existing validatedodels, and the combined model intelligently chooses between
hem.We illustrate our approach with two applications where
cclimatization plays a key role. In the reconciliation of wineermentation kinetics, fermenting yeast cells acclimatize its behav-or to the amount of sugar and nitrogen. In the modeling of thesteo-adipo differentiation system, osteo-adipo precursor cellscclimatize to signals favoring specific cell lineages. Both appli-ations show acclimatization changes with direct relevance iniotechnology (winemaking industry) and in Biomedicine (studyf bone mass disorders). From a modeling point of view, thesepplications involve combining complementary models that have
reviously been validated for specific environmental conditions,hrough strong switches (reconciliation of wine fermentationinetics) and coefficient switches (osteo-adipo differentiation). The
cells respond increasing osteoblasts for specific cell signals.
platform we use for simulating these applications is BioRica2, whichis specially adapted to simulate hybrid models, reuse and combinemodels.
2. Basic notions of acclimatization and hybrid systems
2.1. Acclimatization
Acclimatization is the process in which morphological, behav-ioral, physical or biochemical traits are adjusted in response toenvironmental changes (Varela et al., 1974; Watts et al., 1975; Colesand Brown, 2003). Acclimatization occurs within the organism’slifetime, in contrast to adaptation (Williams, 1974), which is anevolutionary process. Acclimatization capacity is related top heno-typic plasticity, that is the degree to which the organisms are ableto acclimatize (Pigliucci et al., 2006).
To illustrate this phenomenon, environmental changes andacclimatization responses for two case studies treated in this arti-cle are shown in Fig. 1. In both of them, nutrient levels play animportant role: organisms need to modify their function in orderto survive with less nutrients (e.g. allostasis: Sterling et al., 1988;McEwen and Wingfield, 2003). As a result, competitive behaviorbetween organisms or species is common when nutrients are scarce(case of fermenting yeast). From a cellular perspective, changesin environmental conditions can cause cellular stress and inducecell death. According to the organism’s phenotypic plasticity, cellsexhibit stress responses ranging from activating signaling path-ways that promote survival to those that result in apoptosis thateliminates damaged cells (Fulda et al., 2010). Cells transmit andreceive signals to stimulate differentiation into one lineage oranother (case of osteo-adipo differentiation). Ideally, these signalsrespond to the organism’s need of forming a specific tissue type.However, pathological states like cancer can also result when sig-nals are anomalously propagated (Oberley et al., 1980; Vermeulenet al., 2008; van der Deen et al., 2011).
When modeling a dynamic system detected in nature, one mustecide what entities to represent, and what factors and interactions
nclude. In continuous models, system variables change continu-usly over time. Differential equations are one (but not the only)ay of representing these models. In discrete models, dynamics
ccurs discretely at distinct instances in time. In this case, the lawsse logical and recurrence relations. In addition, when consideringiological models there are at least two reasons why randomnessust be taken into account: sampling and incomplete informa-
ion (Wilkinson, 2006). Representative experimental data is usedo recreate reality, but this statistical decision usually causes samp-ing errors. Since obtaining biological data can be expensive andecause of the intrinsically unknown nature of some phenomena,on-deterministic and stochastic models allow multiple behaviorecisions when biological information is incomplete. A model istochastic when it is possible to obtain different answers for theame factor values. A model is non-deterministic if such differentnswers cannot be controlled (Wilkinson, 2006).
Hybrid systems are needed for modeling, since they allowesearchers to combine processes that use different model typesepending on, for instance, the diversity of process componentsr the effect of environmental conditions such as nutrient levelsr temperature. The latter can change the system’s behavior lawsue to acclimatization (Watts et al., 1975; Coles and Brown, 2003).ybrid systems integrate both continuous and discrete dynamics,nd allow stochastic transitions.
A classical hybrid system is formalized as follows. One con-iders two types of continuous variables: the state variables x =x1, . . ., xn) ∈ Rn and the control variables u = (u1, . . ., uk) ∈ Rk; anddiscrete mode variable mode taking finitely many values in a set
1, . . ., M}. The dynamics of the mode variable occurs on discretentervals of time following a deterministic, non-deterministic, ortochastic protocol. Continuous variables evolve according to con-inuous modeling, usually a system of differential equations, andode changes can transform the definition of the continuous mod-
ls. Mode changes can also be induced by special conditions onhe continuous variables, called guards (Henzinger, 1996). This for-
alization is known as a Switched system (Branicky, 1994; Shortent al., 2007). From now on we denote the resulting hybrid modely HYBRID (x,u, mode) and the resulting continuous model whenode = i by BEHAVi(x, u). The domain of HYBRID(x, u, mode) is the
et � ⊂ Rn × Rk × {1, . . ., M} in which the model is valid. If at anyime (x, u, mode) takes a value which does not belong to � then theynamics is finished. More details are provided in next section.
. Hybrid system approach to model acclimatization
Our goal is to provide an adequate modeling protocol thatescribes the dynamic behavior of biological systems includingcclimatization. The modeling protocol we present here is based onhe following premise: acclimatization behavior of biological sys-ems can be directly translated into hybrid systems modeling onceifferent regimes of the system are available. Indeed, this trans-
ation is stated by associating the regimes of the system with itsehavior laws, and defining the conditions of mode changes (guards)s the special environmental conditions inducing the acclimatiza-ion.
In the case of biological systems exposed to acclimatization,ne needs to be able to describe hybrid systems with various lev-
ls of model changes, without losing the capacity to simulate thelobal behavior. An important difficulty is how to model the effectf the mode changes over the continuous dynamics when thesehanges can affect it drastically. In this study, we define and use two
121 (2014) 43–53 45
modeling schemes depending on the level of change in the contin-uous model, coefficient switches and strong switches, that take intoaccount these challenges. Hybrid systems with coefficient switchescorrespond to the so-called Switched systems, with general contin-uous dynamics (Branicky, 1994; Shorten et al., 2007). The notionof strong switches is the contribution we present here taking intoaccount the full complexity of acclimatization.
To use one or another scheme is a modeling decision. Let us sup-pose, for example, a set of differential equations in which, given aspecific regime, the derivative of the variable xi responds linearlywith respect to variable xj. However, passing to a different regime,this dependence becomes quadratic. Therefore, the decision lies inwhether to join the models by another linear combination of bothdependencies and choose the coefficients in function of the con-ditions, or by considering them separately. If one decides to joindifferent dynamic models by using coefficient switches, the mod-els must be rewritten to represent them within a unique structure,possibly by identifying dependency types and grouping them. Ingeneral, however, rewriting is not always the best option. Rewrit-ing models can help to interpret results, but is time consuming,increases the possibility of errors, and often is not meaningful. Ifthe costs of rewriting are too high, due to different type of modelsor implementations or computational complexities, is convenientto use strong switches. Moreover, if one does not have access toa particular model, but can access the state variables values com-puted by this model (e.g. by executing software), one cannot rewritethe model. Such models are seen as black boxes with differentstructures, and thus the system is better modeled using the strongswitches protocol.
As argued in Uhrmacher et al. (2005) and Maus et al. (2008),software adapted to the important needs of reusing and reconcilingmodels must allow both modeling continuous dynamics in a mod-ular way and hierarchical integration of a combination of models.A common specification of biological models is SBML (Hucka et al.,2010), the most popular notation for biochemical reactions modelsgoverned by temporal differential equations. However, simulationsoftwares for SBML-specified models are not yet completely capa-ble of combining models without ambiguities. This represents achallenge because human understanding of biology is modular(Hartwell et al., 1999): biological functions result from interactingprocesses and, through modeling, one associates a model to eachmodule. Examples of functional modules are those for protein syn-thesis, DNA replication, glycolysis and other metabolic pathways(Rives and Galitski, 2003; Marucci et al., 2010). Thus, the model-ing and simulation software must allow connecting models thatseparately describe the dynamics of the state and the control vari-ables, or model separately some of these variables. If models canbe combined in a non-ambiguous way, then one gains the power toimprove and extend them, and thus obtaining increasingly morecomplete descriptions of biological processes depending on thelevel of details desired and the quantity of available information.In the absence of information, one can build models with openspecifications by defining stochastic or non-determinist modules.
As we will address in the following sections, our approach allowsreusing models and, through strong switched hybrid systems, mod-els describing a system at different regimes can be combined in amore general biological model without the need of rewriting them.Thus, the application range of hybrid systems is greatly increased.The fact that models can be completely different is not an imped-iment to combine them in a general model, thus allowing themodeling of even drastic changes in system’s behavior.
3.1. Translating acclimatization into hybrid systems
To translate an acclimatizing process into a hybrid systemwe begin by identifying its possible regimes, according to the
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elationship between the different environmental scenarios andnternal factors where the biological system can live. The set ofegimes is enumerated and we associate a continuous model,escribing the dynamics of state and control variables, to eachegime. Thus, changes in the mode variable model transitionsetween regimes.
.2. A modeling scheme: strong switches
The transition of one regime i to other regime j can dramaticallyhange the dynamic behavior of a system. How one can translatehis into a model? We distinguish two scenarios of change. On oneand the dynamics defined in BEHAVi(x, u) and BEHAVj(x, u) differnly in the parameters of the models, on the other, the completetructure of the dynamics is affected and one can not simulate ity changing parameters. An acclimatization process induces oner more scenario of the former type. In the latter scenario we meanhat there coexist finitely many dynamic models, in which parame-ers can also change with acclimatization regimes, and furthermorehat drastic changes in the acclimatization will force the change of
odels. We call the first scenario coefficient switches and the secondne strong switches (Fig. 2).
To be precise, in strong switches one has L different structuralynamic models, each one being a coefficient switched hybridodel
YBRID(i)(x(i), u(i), mode(i)) for i ∈ {1, . . ., L}nd with mode(i) ∈ {1, . . ., M(i)}. We consider that each hybrid modelas domain �i ⊂ Rn × Rk × {1, . . ., M(i)}, and that the variables x(i)
nd u(i) are local versions of global variables x and u. All these hybridystems are joined through another mode variable called MODEaking values in P({1, . . ., L}), that is, a subset of {1, . . ., L}. ThisODE variable determines which hybrid systems are active at a
iven moment. The MODE variable can be seen as a vector in {0,}L, where 1 means active and 0 inactive. Ideally, at each time, onlyne hybrid model is active and the value of x and u are the valuesf x(i) and u(i) from the active model. If many models are active thealues of x and u depend on the values of x(i) and u(i) from all thective models. A classical way is to consider the average of such x(i)’snd u(i)’s. In the case of strong switches we define mode = (MODE,ode(1), . . ., mode(L)) and, without loss of generality, one refers to
he hybrid system by HYBRID(x, u, mode).
.3. Expliciting the dynamics
We will explain the dynamics of the mode variables MODE andode(i) for i ∈ {1, . . ., L} in the strong switches case. To start we
onsider a hybrid system with coefficient switches, whose modeariable is mode.
ig. 2. Coefficient switches and strong switches. For coefficient switches, mode transitionodel. In a more complex context, the strong switches approach uses a completely diffe
alue 1) from an active submodel. Each submodel has classical mode switches. The generrst two in the scheme).
121 (2014) 43–53
To model the dynamics of the changes to mode we decide, atany time, its next value depending on its present value and thevalues of x and u. One has a finite set of actions, which are subsetsA ⊂ Rn × Rk × {1, . . ., M}. When variable (x, u, mode) belongs to oneof such As, a transition in the mode variable will occur. One canthink of these sets A as those situations determining the change ofthe regime when a biological system will acclimatize. We will usethe boolean variable GA(x, u, mode) to indicate if (x, u, mode) ∈ A.Such functions are called guards.
The actions are not necessarily disjoint. Given the values of x, uand mode, finitely many actions can be possible. We denote such aset as A(x, u, mode) = {A : GA(x, u, mode) = TRUE}. Thus, the dynamicsmust choose one action among all possible ones. We choose theaction through an stochastic criterion described in Eq. 1. In addi-tion, having chosen an action A, a finite number of mode values arepossible. This last non-determinism, when we simulate the system,is solved by a randomized scheduler (De Alfaro, 1998). See Fig. 3summarizing the dynamics of the system.
Let us describe a classical stochastic procedure to choose theaction. Denote by P(A|(x, u, mode)) the probability of choosing theaction A given the values of x, u and mode. We compute this probab-ility as explained in Eq. (1), in which wA ≥ 0 is a weight assigned tothe action A:
P(A|(x, u, mode)) ={ wA∑
B∈A(x,u,mode)wBif A ∈ A(x, u, mode)
0 if A /∈ A(x, u, mode)(1)
After choosing the action A, the function time((x, u, mode), A)measures the delay of that action. It is a random variable with someprobability distribution. During the time determined by time((x, u,mode), A), there is no action possible and the values of x, u andmode remain constant. After the delay, the value of mode changesaccording to the chosen action, and the dynamics of x and u con-tinues according to BEHAVmode(x, u) using the new mode until thenext action.
Now, we consider a strong switched hybrid system HYBRID(x,u, mode). Here we are interested in the dynamics of the globalvariables (x, u) but we need to consider L different hybrid sys-tems HYBRID(i)(x(i), u(i), mode(i)), each one being valid for its domain�i ⊂ Rn × Rk × {1, . . ., M(i)}. In this case, the dynamics of eachmode(i) (for i ∈ {1, . . . L}) is modeled as we described for coefficientswitches. We impose that the dynamics in each module remainsconstant if it is not active by the global variable MODE. As in thecase of coefficient switches, the dynamics of the variable MODE
depends on a set of actions. We consider actions A ⊂ Rn × Rk ×P({1, . . ., L}), and the guard functions HA(x, u, MODE) indicatingwhether (x, u, MODE) ∈ A. Thus, at any time, the chosen action Adecides which HYBRID(i)(x(i), u(i), mode(i)) models for i ∈ {1, . . ., L}
s provoke model changes just by changing values of coefficients in the continuousrent model structure by activating the associated coordinate of MODE (taking theal value of x and u is determined from the local values at the active submodels (the
R. Assar et al. / BioSystems 121 (2014) 43–53 47
Fig. 3. Scheme of the dynamics of a hybrid system with strong switches. We show only one submodel and MODE refers to the component associated to the activity of the localm al actu me(x(td d afte
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continuous, stochastic and non-deterministic behaviors in a non-ambiguous manner allowing multi-scale dynamics, combination ofmodels by input-output connections and action synchronizations,
odel HYBRID(1)(x(1), u(1), mode(1)), which becomes 1 at time t0. The delays of the loc(1)(t3), mode(1)(t3)) at t3 respectively. The delay of the global action A2 is �2 = tiisactivated by the action A2 at the global level, (t3 + �3 > t2 + �2). Before time t0 an
re active depending on the global values of x and u, and the presentalue of MODE.
These two levels of mode changes, the global level associatedith the variable MODE and the local level associated with the
ariables mode(i), which must be consistent. We consider that, hav-ng chosen an action at the global level and while the change inhe value of MODE is not effective (due to its delay), each sub-
odel HYBRID(i)(x(i), u(i), mode(i)) continues according to its hybridynamics. Thus, only when the value of MODE is changed the sub-odels are activated or disactivated according to the new value
f MODE. At the local level, since time variables are independent,he probability of having simultaneous actions is zero. If many sub-
odels undergo local actions at the same time, all such modelsre programmed to execute (after their own delays time((x(i), u(i),ode(i)), A)) the respective mode change. If a change in mode(i) of
he submodel HYBRID(i)(x(i), u(i), mode(i)) was programmed but dur-ng its delay a higher level change in MODE is executed disactivatinghis submodel, then the submodel does not execute the local modehange.
A final important point to explain is how to manipulate thenitial conditions. Given a hybrid system with strong switchesYBRID(x, u, mode), the system is supplied with initial values x =
0 ∈ Rn, u = u0 ∈ Rk, x(i) = x0 ∈ Rn, u(i) = u0 ∈ Rk, mode(i)0 for i ∈ {1,
. ., L}, and MODE0. The submodel HYBRID(i)(x(i), u(i), mode(i)) iseinitialized each time it is activated: the submodel restarts with(i) = x, u(i) = u and mode(i) depending on (x, u).
.4. Reconciliation of models: a theoretical application
When modeling a system, not necessarily in biology, one wantso build a single model that accurately describes the system’s
ynamics under the most general conditions. However, in practicene has several complementary models that describe the dynamicsnder different regimes. The problem is thus to combine such mod-ls in one larger one, that allows one to study the global dynamics of
ions A1 and A3 are �1 = time(x(1)(t1), u(1)(t1), mode(1)(t1)) at t1 and �3 = time(x(1)(t3),2), u(t2), MODE(t2)) at t2. The action A3 is not executed because the submodel isr t2 + �2, the global variables x and u depend on other submodels being active.
the system. We claim that hybrid systems with the strong switchesformalism is very well adapted to this problem.
Let us be more formal. Given a system with state variables x ∈ Rn
and control variables u ∈ Rk, and given coefficient switched hybridmodels HYBRID(i)(x(i), u(i), mode(i)) for i ∈ {1, . . ., L} that describethe system at each given regime �i ⊂ Rn × Rk × {1, . . ., M(i)}, wecombine them in a strong switched hybrid model in the followingway. We call reconciliation of these models the construction of astrong switched hybrid model with domain
⋃Li=1�i where each
module corresponds to HYBRID(i)(x(i), u(i), mode(i)), and the variableMODE choses deterministically each component of MODE; the i-thcomponent of MODE is 1 if the global guard (x, u, mode(i)) ∈ �i. Notethat this definition of MODE means that each submodel is active onits domain. If the intersection
⋂i∈1,. . .,L�i of the domains of active
models at a given time is not disjoint, x and u are computed as theiraverage values in the intersection. There are no delays in globalactions, and we assume that if two models could be joined in acoefficient switched hybrid model then it was already done.
Example 4.1 illustrates a reconciliation of models in a strongswitched hybrid model that describes acclimatization to differentregimes.
3.5. Implementing and simulating the hybrid system dynamics
The implementation of models using hybrid systems forbiological processes under acclimatization is possible througha well adapted software: BioRica3. BioRica (Assar and Sherman,2013) is a high-level modeling framework that integrates discrete,
translation from models in the SBML specification, and hierarchical
elations. This framework has been previously used to simulateybrid systems in (Assar et al., 2012) and also in (Assar andherman, 2013). Here we extend its application to simulate hybridystems with strong switches.
Hybrid systems, in particular with strong switches, could alsoe coded and simulated using other general mathematical soft-are such as Matlab4. The main advantage of using BioRica is that
t defines a specification of models by composition of nodes, each ofhich is structured to declare variables, initial values, the continu-
us model, guards and their effects, as well as to declare submodelss subnodes and combine continuous models by input-output con-ections through the flow field. Using BioRica for strong switches,ach submodel HYBRID(i)(x(i), u(i), mode(i)) is coded by a node. Usingubnodes, we built a main node coding the dynamics of x, u andODE, with input-output connections in which the dependencies
etween the variables x(i), u(i) and mode(i) at each subnode and theariables x, u and MODE at the main node are established. To avoidmbiguities, given a composed node and a variable that is used byany subnodes, one controls how the changes to this variable are
erceived by the sub-nodes by associating to that variable a flowariable, and imposing assertions of equality between the versionsf that flow variable on different nodes.
. Two case studies using the strong switches approach toescribe acclimatization
We analyze two case studies of biotechnological and biomedicalmportance: wine fermentation kinetics and mammal osteo-adipoifferentiation. We describe the modeling protocol and explain inore details each application and the subjacent acclimatization
rocess. We see that determined wine fermentation kinetics isest modeled with strong switches while osteo-adipo differentia-ion can be modeled with coefficient switches but reusing separate
odels for the dynamics of the state and control variables.Both application fields, winemaking industry and bone mass
isorder treatments, are highly relevant. The winemaking indus-ry produces around 60 million hectolitres per year only in France,nd the worldwide loses from stuck and sluggish fermentationsre estimated at 7 billion euros annually. Osteoporosis affects onehird of women and one twelfth of men over 50 years old, andhe advances in life expectancy urge us to look for treatmentshat are effective and with less side effects, therefore improvingife quality. From a modeling point of view, we consider that both
odels cover a wide range of applications, including systems ofifferential equations and Flux Balance Analysis, approaches thatre common in Biological Modeling. These applications combineeparately validated components, taking decisions about how toeuse and connect them. In the case of wine fermentation kinetics,hree a priori defined models are reused without rewriting withtrong switches. It is shown as a particular example of reconcili-tion of models. Although the oste-adipo differentiation model isoefficient switched, the complete model considers a multiple com-ination of different types of models to define the ways and times inhich the stimulatory signals affect the differentiation dynamics.
hus, the model integrates gene regulations and metabolic signalseusing models for Wnt signaling pathway activation and apopto-is.
.1. Wine fermentation kinetics: an application of strong switches
The production of ethanol is the direct result of yeast metaboliz-ng sugar in anaerobic conditions. This pathway is preferred even
Fig. 4. Reconciled model of wine fermentation kinetics. Hybrid system with strongswitches, in which the Coleman and Scaglia models are joined in a hybrid systemwith coefficient switches, and the Pizarro model is a separate module.
when oxygen is available, regardless of the fact that this alterna-tive is energetically more expensive (Bisson, 1999). To understandthe behavior of these fermenting yeasts many efforts have beenundertaken by integrating genomic, proteomic and metabolomicstudies; (see Cherry et al. (2012)) for Saccharomyces cerevisiae inparticular5. Modeling their kinetics may allow us to predict the bestfermentation conditions, as well as to detect and rectify fermen-tation problems. Yeast cells acclimatize to changes in conditions,affecting its kinetics. In particular, the risk of stuck fermentationincreases with high sugar and temperature levels (Bisson, 1999),which is an unfortunate consequence of competition within theyeast population for resources. This behavior change, competition,is an acclimatization to scarce resources when many yeasts haveavailable in low nutrient concentrations (Fig. 1).
To describe this system, we used a strong switched hybrid sys-tem approach. We have previously (Assar et al., 2010) consideredthe dynamics of three models: Coleman et al. (2007), Scaglia et al.(2009) and Pizarro et al. (2007). These models are defined by dif-ferential equations. The main difference between the Coleman andScaglia models is that the former includes temperature as a vari-able, while the latter takes into account competition for availableresources. The Pizarro model combines differential equations withan iterative optimization approach using FBA (Flux Balance Analy-sis, Palsson, 2006; Orth et al., 2010) to define the dynamics of thecontinuous variables. To find the domain of each model, we evalu-ated them by comparing predictions with experimental results ofthree papers: (Pizarro et al., 2007; Malherbe et al., 2004; Mendes-Ferreira et al., 2007). More details about the models and theirstatistical validation can be seen in Assar et al. (2010). In thatvalidation we considered three factors: sugar S, nitrogen N andtemperature T; each one taking level values characterized by realintervals: low (L), moderate (M), and high (H). For each combina-tion of levels at the initial time (S(0), N(0), T(0)) (that we call factorsconfiguration) and separating the dynamics into transient and stableregimes we qualified the accuracy of the models. Changes of regimecan only occur from transient to stable.
The strong switched hybrid model was defined by reconciliationas follows (see Fig. 4):
• The state variables we considered are x = (X, S, E), where X is thebiomass, S is the sugar concentration and E is the ethanol concen-tration (in g/liter).
The control variables are u = (N, T), with N the nitrogen concentra-tion and T the temperature.We considered two modules of coefficient switched hybrid mod-els: HYBRID(1)(x(1), u(1), mode(1)) in which the Coleman andScaglia models were joined, and HYBRID(2)(x(2), u(2), mode(2)) forthe Pizarro model.At the local level, the mode variables mode(1) and mode(2) aretuples of the form 〈(S(0), N(0), T(0)), Model〉, where Model isthe active model. For mode(1) the possible values of Model areColeman, Scaglia, or Neither, if the coefficients are taken respec-tively from the formulas in Coleman, Scaglia, or from neither.For mode(2), there are no changes in mode(2) and Model is alwaysPizarro.According to validation results, HYBRID(1)(x(1), u(1), mode(1)) andHYBRID(2)(x(2), u(2), mode(2)) are valid in the following domainsrespectively (to simplify the notation we denote local versionsby x and u):
�1 = { (x, u, 〈(L, H, H), Scaglia〉) | X ≤ 1}∪{(x, u, 〈(M, M, H), Scaglia〉) | X ≤ 3.5}∪{(x, u, 〈(M, H, H), Scaglia〉)}∪{(x, u, 〈(H, M, H), Coleman〉) | S ≥ 55}∪{(x, u, 〈(H, H, H), Scaglia〉) | S ≥ 30 }
�2 = { (mode(2) = 〈(L, M, H), Pizarro〉)}∪{(x, u, 〈(L, H, H), Pizarro〉)}∪{(x, u, 〈(M, M, H), Pizarro〉)}∪{(x, u, 〈(H, M, M), Pizarro〉)}∪{(x, u, 〈(H, M, H), Pizarro〉) | S < 55}∪{(x, u, 〈(H, H, H), Pizarro〉) }
At the global level, MODE is a tuple of the form
〈(S(0), N(0), T(0)), (i, j)〉 ,
where i = 1 if HYBRID(1)(x(1), u(1), mode(1)) is active and i = 0 if it isinactive, j = 1 if HYBRID(2)(x(2), u(2), mode(2)) is active and j = 0 if itis inactive.
ig. 5. The dynamics of the mode variables MODE and mode(1). Both variables share theonfiguration, when passing to the stable regime both values can change, and finally they), −) for mode(1), do not have changes (they start in the stable regime).
121 (2014) 43–53 49
• The local and global actions modify the values of the last coor-dinates (i, j) of mode(1) and MODE respectively. The actions aredeterministic, without delays and, in the case when two modulesare active, values of x and u are average values (see Section 3.4).Each mode variable suffers at most one change during its dynam-ics. Such a change can happen when the dynamics passes from thetransient to stable regime. The guards are defined using thresholdfunctions over the coordinates of the variable x, which character-ize whether the dynamics is in stable or in transient regime (seeFig. 5).
Let us describe through two examples the information about ini-tial values and guard conditions in Fig. 5. If the factors configurationis (H, H, H) then initially at the global level only the model in the sec-ond module (Pizarro model) HYBRID(2)(x(2), u(2), mode(2)) is active(MODE = 〈(H, H, H), (0, 1)〉), and at the local level mode(1) is preparedto take the coefficient values with the Scaglia formulas if the firstmodule HYBRID(1)(x(1), u(1), mode(1)) is activated (mode(1) = 〈(H, H,H), Scaglia〉). This last occurs when S < 30 and the second moduleremains active (MODE = 〈(H, H, H), (1, 1)〉). If the factors config-uration is (M, H, H), at the initial condition only HYBRID(1)(x(1),u(1), mode(1)) is active (MODE = 〈(M, H, H), (1, 0)〉) and computescoefficient values with the Scaglia formulas (mode(1) = 〈(M, H, H),Scaglia〉), and when S < 50 the coefficients of HYBRID(1)(x(1), u(1),mode(1)) change to compute coefficients with the Coleman formulas(mode(1) = 〈(M, H, H), Coleman〉).
In simulations shown in Fig. 6 (which illustrates sugar con-centration starting with the (H, H, H) configuration) we considerthat the initial values of the mode variables mode(1), mode(2),and MODE are 〈(H, H, H), Scaglia〉, 〈(H, H, H), Pizarro〉, and 〈(H,H, H), (0, 1)〉 respectively. Reconciliation improves the dynamicsprediction at both qualitative and quantitative levels. It qualita-tively agrees with the expected values for high levels of sugarand temperature: initially, the nutrients are abundant and com-petition is non-existent, but competition appears (with the Scagliamodel) when the resources become scarce. As shown in Fig. 6, the
activation of the Scaglia model also quantitatively improves thepredictions at the stable regime. This model could be improvedby adding more factors, such as pH and phenotype characteristicsof the culture (yeast mortality and senescence), or by considering
same dynamics scheme, at the transient regime the value is linked to the factorsremain constant. Some mode values, such as ((L, M, H), (0, 1)) for MODE and ((L, M,
50 R. Assar et al. / BioSystems
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ig. 6. Result for sugar concentration with the (H, H, H) initial configuration. Thehange from transient to stable phase provokes competition. This is described byhe Scaglia model which becomes active.
ore classification levels and models to reconcile. We show resultsor other configurations in Supp. Material.
As shown in (Chen et al., 2010), osteoblasts and adipocytes shareommon precursor cells, derived from bone marrow stromal cells.uring the differentiation process, every precursor cell, called pro-enitor, can proliferate, differentiate into either an osteoblast or andipocyte lineage, or do apoptosis, depending on stimulus signals.hus, the acclimatization of the osteo-adipo system consists in thectivation of cell lineages or modification of death rates in responseo specific stimulus signals. An accurate model of this osteo-adipoifferentiation system would allow testing and calibrating in silicohe combination of stimulatory signals (Assar et al., 2012). This inurn may bring a useful mathematical tool to simulate biologicalesponses to new drugs under experimental consideration to treatsteoporosis and other bone mass disorder with less side effects.
To describe this system we use a coefficient switched hybridystem approach (Fig. 7). The dynamics of cell fate decisions whenoing from osteo-adipo progenitor cells to bone (osteoblast) or fatadipocyte) cells can be modeled as a coefficient switched hybrid
odel, where osteoblast or adipocyte lineages are favored depend-ng on external stimuli (Fig. 1, Assar et al. (2012)). This model canredict changes in bone and fat formation by either stimulating or
nhibiting the Wnt/ˇ-catenin pathway (Kim et al., 2007) (associ-
ted with the osteoblast lineage), the PPAR� pathway (associatedith the adipocyte lineage), the division of progenitor cells, and
poptosis of progenitor and osteoblast cells by increasing the lev-ls of homocysteine (Kim et al., 2006). The expression of progenitor
ig. 7. Hybrid system of osteo-adipo differentiation system. Modeling scheme with coeentration of progenitors, osteoblasts and adipocytes respectively. The dynamics of u =ˇesponds to stimulus signals to each lineage and to modify the apoptosis rates.
121 (2014) 43–53
biomarkers (e.g. OCT4, SOX2 or both, MacArthur et al. (2008)) isassociated with maintenance in the uncommitted state, which pre-vents the expression of RUNX2 (runt-related transcription factor 2(Krishnan et al., 2006) and PPAR� (peroxisome profilerator-activatedreceptor gamma (Chen et al., 2010) and subsequent engagementto osteogenic and adipogenic differentiation, respectively. Theinteraction of these three cell lineages is established as a Gene Reg-ulatory Network (de Jong, 2002), modeled by a system of non-lineardifferential equations, continuous model which is the central partof the model.
The coefficient switched hybrid model HYBRID(x, u, mode) isdefined as follows (see Fig. 7):
• The state variables we consider are x = (xP, xO, xA), denoting theconcentration of progenitors, osteoblasts and adipocytes respec-tively.
• The control variable is u, the concentration of the nuclearˇ-catenin–TCF complex indicating activation of the Wntpathway.
• The continuous dynamics is modeled by two models: the differ-entiation dynamics described by differential equations with statevariables x and the model reused from (Kim et al., 2007) describ-ing the dynamics of u.
• The mode variable mode consists in five components controllingparameters of the differential equations:– zD taking value 1 if the differentiation is stimulated and 0 if it
is not,– zO taking value 0.8 if the formation of osteoblasts is stimulated
(the Wnt/ˇ-catenin pathway active) and 0 if it is not,– zA with value 0.8 if the formation of adipocytes is stimulated
and 0 if it is not,– kP (the decay rate of progenitors) with value 0.141 if the apo-
ptosis of progenitors is stimulated (high levels of homocysteine)and 0.1 if it is not,
– kO (the decay rate of osteoblasts) taking value 0.441 if the apo-ptosis of osteoblasts is stimulated (high levels of homocysteine)and 0.3 if it is not.The values of the mode variable mode modify the equilibrium
states to favor one or other cell lineage. In particular the valuestaken by zD, zO, zA at each regime, stimulated or not each lineage,were obtained from previous works by Schittler et al. (2010). Thevalues for decay rates kP and kO were obtained by calibratingresults to human bone cells (Assar et al., 2012) and observationsby Kim et al. (2006).
• The actions represent the activation of stimulatory signals. Weconsider 5 actions that separately modify the value of each com-
ponent of mode. The guard condition u > 1.1 · un (with constantun = 8.81) indicates that the Wnt pathway is active and the com-ponent zD of mode changes its value to 0.8 deterministically andwithout delay. The actions affecting the other components of
fficient switches in which xP , xO and xA are the state variables describing the con--catenin–TCF is modeled according to (Kim et al., 2007). The mode variable mode
R. Assar et al. / BioSystems
Fmc
Fcs
ig. 8. The guards, delays and the effects on each component of the mode variableode. Each guard acts on a specific component of mode (we only show the affected
omponents), which simulate an specific stimulus.
mode have stochastic delays given by exponential distributions.The details of guard conditions and effects in each mode compo-nent are explained in Fig. 8. Thus, similar to that imposed in thewine fermentation example, each mode component can change at
ig. 9. Dynamics of xP (D.progenitors), xO (D.osteoblasts) and xA (D.adipocytes). Con-entration of progenitors, osteoblasts and adipocytes, and the effect of stimulusignals.
121 (2014) 43–53 51
most one time and there is no option to come back to the previousvalue.
With this hybrid model, we can obtain the effect of all the con-sidered stimulus signals in the formation of the cell lineages. Inparticular, in Fig. 9, we consider that the initial value of the modevariable mode is (0, 0, 0, 0.1, 0.3), that is to say, the system startswithout stimulus to differentiation, formation of osteoblasts andadipocytes, and apoptosis. We can observe over time how initiallythe stimulation of the differentiation increases the transformationof progenitor cells into osteoblasts and adipocytes, how the activa-tion of PPAR� stimulates the formation of adipocytes and inhibitsosteoblasts, how the activation of the Wnt pathway stimulatesthe formation of osteoblasts, and finally how the apoptosis stimulireduce the number of osteoblasts and progenitors.
5. Conclusions and discussion
Acclimatization describes how changes in the environmentaffect the behavior of organisms. It often involves a significantinvestment by the organism, in ways that prevent it from easilyreturning to the previous behavior. In this work we have pro-posed a methodological approach for incorporating acclimatizationphenomena in order to build a more complete model of a biolog-ical system. We can incorporate a range of changes, from simpleadjustments to an existing model, to radical changes that requirecompletely different models of behavior.
We use hybrid systems theory and reconciliation to presentschemes capable of modeling different levels of changes in thebehavior laws of the system. Continuous models describe thedynamics of system variables with gradual changes over time,while switches are used to change behavior laws and or combinedifferent models. Changes in the continuous dynamics of the sys-tem are generated by mode transitions switching the continuousmodel, where mode transitions can be caused by specific conditionsof the continuous variables. In function of the mode transitions,one transforms the continuous model by changing only the valuesof coefficients (coefficient switches), or by modifying strongly themodel (strong switches). We extend the previous used notions inhybrid systems theory, allowing different levels of model changesand the reconciliation of models. We simulate hybrid systems withthese characteristics using the BioRica framework, which is spe-cially adapted to hybrid systems with strong switches.
We illustrate this methodology in two applications. In acclima-tization of winemaking yeasts to fermentation conditions, wereconcile the three wine fermentation kinetic models of Colemanet al. (2007), Scaglia et al. (2009) and Pizarro et al. (2007), topredict fermentation problems under 12 different regimes. In par-ticular for the HHH configuration first the nutrients are abundantand competition does not exist, but as nutrients become scarcecompetition and the risk of stuck fermentation increase. In acclima-tization for osteo-adipo differentiation, we model how stimulussignals promote progenitor cells to osteoblast or adipocyte lineagecommitment. With our approach, one can consider a coefficientswitched hybrid system with many stimulus models, which couldcontribute to in vitro analysis of the physiological response to bonemass disorder treatments.
There are many other biological systems in which we can con-sider hybrid systems and acclimatization. We previously showed(Assar and Sherman, 2013) a number of applications of hybrid sys-tems, with coefficient switches and strong switches, in Biology
but also in other fields (see the car, the radiator, and the rockinghorse). As seen in the case of the osteo-adipo model, Gene Reg-ulatory Networks can be modeled by hybrid systems with modelcoefficients switched by system conditions. Even in a more basic
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evel, if a gene regulatory network is modeled by a system of dif-erential equations, the simplification of Hill functions (describingctivation/inhibition dynamics) by step or by logged functions gen-rates also a hybrid system (see de Jong, 2002; de Jong et al.,003). We can find acclimatization in the dynamics of organismommunities, each time a substrat (nutrient or specie) varies sig-ificantly. A very well-known way to mathematically describeuch community dynamics is by the Lotka–Volterra like models,eing the simplest example the predator–prey system. Accordingo the values of coefficient parameters, and also initial conditions,he Lotka–Volterra model evolves to convivence or extinction ofpecies. Thus, changes in the parameters of interaction betweenpecies can model acclimatization responses which, depending onhe regime, point to more cooperative or more competitive behav-ors (Lotka, 1956; Selgrade, 1989; Vitanov and Dimitrova, 2013).n example we are particularly interested is the dynamics of com-unities of bioleaching bacteria, which work a key role in copperining allowing copper extraction from deposits with low cop-
er grade. It has been observed that the dynamics of bioleachingacteria varies importantly depending on sulfur and ferric sources,s well as pH and ecosystem composition (Bobadilla Fazzini et al.,013; Martınez et al., 2013; Holmes et al., 2009; Valdés et al., 2008).
The proposed modeling approach allows one to define a globaliew of the dynamic behavior of a system by incorporating dif-erent regimes, modeled independently, in a single model. Ourpproach thus promotes reusability and unambiguous combina-ion of complementary models, and provides a basis for practicalybrid modeling of complex biological systems.
cknowledgements
This work was supported by FONDAP 15090007, Basal grantMM PFB-03, Fondecyt 3130762 and Project CIRIC-INRIA Chile.
ppendix A. Supplementary Data
Supplementary data associated with this article can be found,n the online version, at http://dx.doi.org/10.1016/j.biosystems.014.05.007.
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