This paper is concerned with the control problem of biological systems modeled with Timed Continuous Petri Nets under in nite server semantics. This work introduces two main contributions. The rst one is a bottom-up modeling methodology that uses T CP N to represent cell metabolism. The second contribution is the control wich solves the Regulation Control Problem (RCP ) (to reach a required state and maintain it). The control is based on a Lyapunov criterion that ensures reaching the required state.
Petri nets P N [1], [2], [3] are a formal paradigm for modelling and analysis of systems that can be seen as discrete dynamical systems. Unfortunately, due to state explosion problem, most of the analysis techniques cannot be applied in heavy marked Petri nets. In order to overcome this problem, the Petri net community developed the Timed Continuous Petri Nets (T CP N ) [4], [5], a relaxation of the Petri Nets where the marking becomes continuous and the state equation is represented by a positive, bounded set of linear di¤erential equations.
The main T CP N characteristics such as the nice pictorially representation, the mathematical background, the synchronization of several products to start an activity and the representation of causal relationship make T CP N amenable to represent biochemical reactions and cell metabolism. In fact T CP N marking captures the concentration of molecular species while di¤erential equations together with the ring vectors represent the reaction velocity and the graph captures the metabolic pathways. The entire T CP N captures the cell metabolome.
Several works model [6], [7], analyse [8], [9] and control [10], [11] metabolic pathways. Most of them deal with pseudo-steady states of the biochemical reaction dynamic. Nowadays, the scienti c community is exploring the use of P N and their extensions [12], [13] to model biological systems since the former are able to capture the compounds ow, the reaction velocity, the enabling/inhibiting reactions and both the transitory and steady states of reaction dynamic into a single formalism.
This work is concerned on how to model the entire metabolome with T CP N . It proposes a bottom-up modeling methodology where biochemical reactions are modeled through elementary modules, and shows how these modules are merged to form metabolic pathways, and at the end the cell metabolism. The resulting model captures both, the transitory and steady state metabolome dynamics. It is worth noticing that the derived T CP N model condenses several particular behaviors represented by the set of di¤erential equations generated by the T CP N itself. For instance, a single transition with four input places (a reaction needing four substrates) generates a set of four possible di¤erential equations while two transitions with four input places each will generate a set of sixteen possible di¤erential equations. Therefore highly complex behaviors emerging from few compounds interacting can be captured by T CP N .
This work also presents the control problem of reaching a required state (marking) representing a certain metabolite concentration. In order to solve this problem, an error equation is stated and stabilized using a Lyapunov approach. The solution is the reaction rate vector which is greater or equal to zero and lower or equal to the maximum settled by the kinetics of Michaelis-Menten for the current enzyme concentration. Thus, if a solution exists, it could be implemented in vivo by directed genetic mutation, knock-in (or knock-out) strategies or pharmacological e¤ects.
Present paper is organized as follows. Section 2 gives T CP N basic de nitions, controllability and cell metabolic concepts. Next section introduces the proposed metabolome modeling methodology. Section 4 presents the problem of reaching a required state and synthesizes Lyapunov like transition ow for solving this problem. Following section presents an illustrative example to show the performance of the computed control law. In the last section the conclusions and future work are presented. 2
Basic De nitions This section presents briey the basic concepts related with P N , Continuous P N and T CP N . An interested reader can review [3], [14], [15] and [16] for further information. At the end of this section a useful form of the state equation for T CP N under in nite server semantics is presented. 2.1
De
nition 1. A Continuous Petri Net (ContP N ) system is a pair (N; m0),
where N = (P; T; P re; P ost) is a Petri net structure (P N ) and m0 2 fR+ [0gjP j
is the initial marking. P = fp1; :::; png and T = ft1; :::; tkg are
nite sets of
elements named places and transitions, respectively. P re; P ost 2 fN [ 0gjP j jT j
(
Timed continuous Petri nets De nition 3. A timed ContP N is the 3-tuple T CP N = (N; ; m0) ; where N is a ContP N , : T ! fR+gjT j is a function that associates a maximum ring rate to each transition, and m0 is the initial marking of the net N . are the Pre and Post incidence matrices, respectively, where P re[i; j], P ost[i; j] represent the weights of the arcs from pi to tj and from tj to pi, respectively. The Incidence matrix denoted by C is de ned by C = P ost P re:
Each place pi has a marking denoted by mi 2 fR+ [ 0g. The set ti = fpj j P re[j; i] > 0g ; (ti = fpj j P ost[j; i] > 0g) is the preset (postset) of ti: Similarly the set pi = ftj j P ost[i; j] > 0g ; (pi = ftj j P ost[i; j] > 0g) is the preset (postset) of pi.
A transition tj 2 T is enabled at marking m i¤ 8pi 2 tj , mi > 0. Its enabling degree is: enab(tj ; m) = min mi
pi2 tj P re [i; j]
and it is said that mi constraints the
ring of tj . Equation (
m( ) = Cf ( ) where f ( ) = ( ) And under the in nite server semantics, the ow of transition tj is given by fj ( ) = j enab(tj ; m( )) where j represents the maximum ring rate of transition tj . Notice that T CP N under in nite server semantics is a piecewise linear system (a class of hybrid systems) due to the minimum operator that appears in the enabling function of the ow de nition. De nition 4. A con guration of a T CP N at m is a set of (p; t) arcs describing the e¤ ective ow of all transitions.
(m) [i; j] =
1 Pre[i;j] if pi is constraining tj
0 otherwise De nition 5. The maximum ring rate matrix is denoted by
According to previous notation, the state equation and the ow vector are described by: = diag
1; : : : ; jT j : m = C f =
The only action that can be applied to a T CP N system is to slow down the
ring ow. The forced ow of a controlled transition ti becomes fi ui where
fi is the ow of the unforced system (i.e. without control) and u is the control
action, with 0 ui fi. The controlled state equation is:
(
Ici
1. 2.3
Controllability The classical linear systems de nition of controllability cannot be applied to T CP N systems because the required hypothesis are not ful lled, that is, the input should be unbounded and the state space should be RjP j. The next de nitions are taken from [18]. De nition 6. Let N be net of a T CP N . The structural admissible states set is de ned as SASS (N ) = fR+ [ f0ggjP j (all inital markings that can be imposed to a net). Let B be the base of the left annuller of the incidence matrix C. The equivalence relation : SASS (N ) ! SASS (N ) is de ned as m1 m2 i¤ BT m1 = BT m2, 8m1; m2 2 SASS(N ). The system admissible states set is the equivalent class of the initial marking Class (m0) under .
In the sequel, let us denote by int (Class (m0)) the set of relative interior of
Class (m0).
De
nition 7. Let (N; ; m0) be a T CP N system. It is fully controllable with
bounded input (BIF C) if there is an input such that for any two markings
m1; m2 2 Class (m0), it is possible to transfer the marking from m1 to m2
in
nite or in
nite time, and the input ful
lls (
An equilibrium point represents a state in which the system can be maintained using the de ned control action. Given an initial marking m0 and a required marking mr, one control problem is to reach mr and then keep it. For a further information about equilibrium points an interested reader can review [19]. 2.4
For the wellbeing of an given organism, each cell of that organism must transform the substances available in its surroundings to useful molecules. Such transformations take place as chemical reactions catalyzed by enzymes. In these reactions, a substrate tightly binds non-covalently to its enzyme active site to build an enzyme-substrate complex. At that moment, the enzyme chemically changes the substrate into one or more products and then releases it. The enzyme did not su¤er any irreversible alterations in the process, and now is free to accept a new substrate [20].
There is no limit to the number of possible reactions occurring in nature. Nonetheless, after exhaustive analysis certain general patterns had emerged that became useful to describe several characteristics of biochemical reactions. In the case where a sole substrate becomes a single product, the reaction process is represented by the scheme:
E + S
ES ! EQ
E + Q
(
Typically, the rate of these reactions is settled by the kinetics of MichaelisMenten [21]. Under this kinetic model, the enzyme and substrate react rapidly to form an enzyme-substrate complex while [S] and [ES] are considered to be at concentration equilibrium (the same applies to [EQ] and [Q]), that is, the rate at which ES dissociates into E + S is much faster than the rate at which ES brakes down to EQ.
Throughout the present work, we will consider a physiological cellular state where [S] >> [E], which means that [S] [ES] equilibrium will always tend to complex formation. Therefore, ES dissociation rate is irrelevant and Scheme 11 can be abbreviated as follows:
E + S ! E + Q where the association-dissociation is implicit.
In reactions with more than one substrate, binding can occur in di¤erent
sequences; for instance, the following scheme represents an enzyme system with
two substrates and all the possible sequences:
8 ES1 + S2
>
>
>
>
<
>
>
>
>: ES2 + S1
(
ES1S2 ! E + Q1 + ::: + Qn
Frequently the product of an enzyme is the substrate of another reaction
and so on, to build a chain of reactions called metabolic pathways represented
by M P j = 1j 2j nj where ij is a reaction (
Then a (Cell) Metabolome is CM = M P i M P i is a metabolic pathway , and the purpose of CM is to produce a particular set of metabolites in certain concentrations, essential to that cell. 3
In order to model the metabolome using T CP N it is necessary to identify how the elements involved in it will be represented. The next table relates the meaning of each element of the T CP N with respect to metabolic reactions.
Place Molecular Species Marking Concentration Transition Reaction Firing Rate Rate of Reaction
Arc Weights Stoichiometric Coe¢ cients
The bottom-up approach herein proposed to model the metabolome consists of: a) representing reactions, the results of this stage are the elementary modules; b) merging elementary modules, where places of elementary modules representing the same molecular species on the same physical space in the cell will merge Petri Nets & Concurrency { 93 into a single place. The results of this stage are pathway modules; and c) merging pathways modules, where places of pathways modules representing the same molecular species on the same cellular space will merge into a single place; the result of this stage is the metabolome model. For stages b and c, any specie being protein-mediated transported into a di¤erent organelle shall be modeled through the same elementary module, representing instead of substrate and product the same molecule in di¤erent spaces.
Next section describes these stages.
In order to represent each reaction i with T CP N elementary modules
representing the Scheme (
Associated to transition ti is i representing the rate of reaction. Example 1. Let P 1 + E1 ! P 2 + E1 be the 1 reaction. There is one place for each molecular species (P 1, P 2 and E1), and one transition t1 representing 1. Finally, arcs are xed in the way depicted in Figure 1a.
Assuming that the substrate concentration will remain higher than the enzyme concentration (this is an expected behavior of the system), the conict between substrate and enzyme can be ignored. Hence, if a system has 2n con gurations originated by the n number of enzymes in conict with substrates, all those con gurations are eliminated because min ([E]; [S]) = [E] for all 0. Let N 1 and N 2 be two elementary modules, then the merging N is such that N = (P; T; P re; P ost) where P = P 1 [ P 2, T = T 1 [ T 2, P re = P re1 [ P re2 and P ost = P ost1 [ P ost2. Notice that places representing the same molecular species in the same physical space are merged into a single place.
After a merging of elementary modules is made, pathway modules are obtained.
Example 2. Let N 1 = P 1; T 1; P re1; P ost1 and N 2 = P 2; T 2; P re2; P ost2 be two elementary modules showed in Figure 1a and Figure 1b respectively. Then, the merging is N = (P; T; P re; P ost) where P = P 1 [P 2 = fP 1; :::; P 5; E1; E2g, T = T 1 [ T 2 = ft1; t2g and arcs are xed in the way depicted in Figure 2, where the merging is showed. 3.3
Let N 1 be a pathway module and N 2 be a pathway or an elementary module, then the merging N is such that N = (P; T; P re; P ost) where P = P 1 [ P 2, T = T 1 [ T 2, P re = P re1 [ P re2 and P ost = P ost1 [ P ost2. Notice that places representing the same molecular species are merged into a single place.
After a merging of pathway modules is made, a metabolic model is obtained. Example 3. Let N 1 = P 1; T 1; P re1; P ost1 be the pathway module showed in Figure 2. Let N 2 = P 2; T 2; P re2; P ost2 and N 3 = P 3; T 3; P re3; P ost3 be two elementary modules showed in Figure 1c and Figure 1d respectively. Then Petri Nets & Concurrency { 95 9 the merging is N = (P; T; P re; P ost), where P = [P i = fP 1; :::; P 5; E1; :::; E4g, T = [T i = ft1; t2; t3; t4g for i = 1; :::; 4. Arcs are xed in Figure 3, where the merging is showed.
Although obtained metabolic models could be not live, the addition of a virtual transition and arcs going from the last place representing nal products to the virtual transition and from virtual transition to the places representing initial products with an appropriate virtual reaction velocity will make the metabolic model live. For instance, consider the net of Figure 1a, it is a non-live net, but if we add a virtual input transition tv to the place S and a virtual output transition tv to the place Q the system will gain liveness, see Figure 4. Notice that tv must to be the same transition added to the inital and nal metabolites, this is because it is necessary to maintain the conservativeness of the matter of the system. This notion is based assuming that each module belongs to a bigger system, therefore, although the real input and output transitions could be not the same, they must have the same ring ratio. 4
An important control problem in the metabolic engineering area is to reach a certain metabolome state such that the production of selected metabolites is regulated or particular processes are limited or favored. This problem is captured in T CP N as the reachability problem, i.e. to reach a required state mr from an initial state m0 by means of an appropriate control action. This is formalized as follows. De nition 9. Let T CP N be a metabolic model. Then the Regulation Control Problem in (mr; Icr ) (RCP (mr; Icr ))deals with the computation of a control law (m) 0jT j jT j ; then the state equation of xT CP N is: mx = " m # ma =
CIc Ic Remark 1. Notice that the extension has the same dynamic over the metabolic model places and the extra places can only increase its marking. In fact, due to the T CP N is live, then by construction the xT CP N is also live. Then there exists at least one enabled transition. Hence (m) m > 0 (or equivalently ma 0; the zero could be forced by an appropriate control law Ic). Example 4. An example of an extended net is presented in Figure 5. 4.1
Theorem 1. Let (N; m0; ) be a metabolic model T CP N and let xT CP N =
(Nx; m0x ; ) be its extension. If (N; m0; ) is BIC over int (Class (m0)) (notice
Petri Nets & Concurrency { 97
(
Now, let
and
where
that neither the initial marking nor the required marking could be zero
components) and (Icr ; mr) is an arbitrary equilibrium point, then there exists Ic( );
0 f feasible in the T CP N such that m( ss) = mr, Ic( ss) = Icr ;
Proof. If the system is BIC over int (Class (m0)), then there exists a positive
solution r( ) feasible such that
mr = m0 + C r
and IjT j is an identity matrix of order jT j jT j. We claim that V (ex) is a
Lyapunov function, i.e it is positive de
nite and its derivative is negative de
nite.
Since Equation (
we obtain: and Thus, from Equation 16 m( f ) = mr; then ex = 0; a contradiction. Hence V (ex) is positive de nite.
Now , we prove that V (ex) is negative de nite. The di¤erentiate of V (ex) is: thus V (ex) < 0 and V (0) = 0:
Since ma(0) = 0 and it only increase its value, then Ici is feasible leading from ma(0) = 0 to ma( f ) = r; i.e. from m0 to mr. Moreover, assuming mr 2 int (Class (m0)) it is reached in nite time because ma [i] = m (min ( ti)) e and m (min ( t)) 6= 0 8 . At f the control law must be switched from Ic( f ) to Ic( ss) = Icr and the regulation control problem is solved.
The solution to the RCP (mr; Icr ) include both, the transitory and steady state control of metabolic systems. It is an improvement to current control solutions, where the biologist and metabolic engineers use stoichiometric nondynamical approaches such as F BA (Flux Balance Analysis) [22], [23], [24] for the control of metabolic systems. Those are based on a pseudo-stationary state model, represented by the equation:
Sv = 0
where S is the matrix of stoichiometry coe¢ cients and the solution v gives the
balance of mass for a single equilibrium point at that state (v is the reaction
rates vector in a steady state).
(
Example 6. In Figure 8 the evolution of marking ma is depicted. When occurs ma[i] = r[i] the control Ici = 0 makes fi = 0 and ma[i] is maintained until = f (ma = r). Then Ic switches to Icr for the steady state control.
Fig. 8. Marking ma of the Example 5. Petri Nets & Concurrency { 101 15 6
This work presented a model methodology to capture the metabolome behavior. It uses a bottom-up approach where each individual biochemical reaction is modeled by elementary T CP N modules and, afterwards, all the modules are merged into a single one to capture the whole metabolome behavior. Such characteristic of the methodology makes it simple and easy to use while the complex cell metabolic behavior is captured. This work also presented the problem of reaching a required metabolome state. The solution to this problem are the instantaneous reaction velocities that are realizable in biological system.
Present results are being applied to optimize metabolome fermentation in the production of tequila and to biofuels generation.
Future perspective involves introduction of stochastic modelling and merging the metabolome with the signaling and genetic networks.