### Metabolic function optimization -

The control function is now optimized in order to obtain a maximum yield of biomass at the end of the run-time t final.

Three different methods, assuming various levels of information about the network, are considered in order to attain this goal. The first method, direct optimization, is used as a benchmark to compare the results of the other methods.

The last two methods rely on a Bi-level optimization and illustrate a possible solution to the optimization problem when the information about the network is incomplete.

The first method, Direct Optimization, is used mainly as a benchmark, to compare the results of the following methods. Since it is assumed that all the information about the network kinetics is known, the system of differential equations, described in 2 is used.

The value of t reg that results on a maximum product yield is then determined by solving a simple optimization problem. The optimization was tested with two MATLAB functions: fmincon , from the standard optimization toolbox, that finds the minimum of a constrained nonlinear multi variable function, and simannealingSB from Systems Biology Toolbox [ 17 ] that performs simulated annealing optimization.

The Bi-Level optimization algorithm was structured so as to accommodate missing information on the network kinetics. The boxed metabolites and fluxes from Figure 1 are a part of the network that might not be fully described in terms of kinetics.

In this approach the missing kinetic information is replaced by stoichiometric data and flux balance analysis is used to obtain the proper flux distribution. Then, an inner optimization determines the fluxes during the batch time.

The first step of the inner optimization process is to define the initial conditions of the input x 1 and outputs x 3 , x 5. A valid distribution for the fluxes v 1 , v 2 , v 3 and v 4 is then obtained. During this time interval the function u t and the values of v 1 , v 2 , v 3 and v 4 are kept constant.

The time interval for the integration was defined to be 1 second. The inner optimization process allows us to obtain the product yield, x 5 t final , given a certain u t , taking into account a valid approximation of the network dynamics over the simulation time.

The detailed fluxogram of the inner-optimization is shown in Figure 6. The bi-level optimization algorithm can be represented schematically as in Figure 7. On the first implementation of the Bi-Level optimization algorithm the dynamics of the boxed metabolites from Figure 1 are used but, following the algorithm structure, steady-state is assumed.

Thus, x ˙ 2 and x ˙ 4 from 2 become:. In this algorithm implementation, the inner optimization problem determines the profile of the metabolites, instead of fluxes, due to the nature of the equations.

The metabolite concentrations are calculated at the beginning of each time interval, solving a Geometric Programming problem, and used with 2 to integrate the values of x 1 , x 3 and x 5 during that interval.

On the second implementation it is assumed that only stoichiometric information is available for the reactions inside the box of Figure 1.

Assuming steady state, the equations of x ˙ 2 and x ˙ 4 become:. Figure 1 shows a regulation from x 3 Biomass to flux v 3. Since stoichiometric models do not account for feedbacks, the effect of x 3 can not be integrated directly in the equations.

Assuming that the forward feedback leads to an over expression of flux v 3 , then a valid solution is to model the forward feedback as a variation of the constraints applied to flux v 3.

Setting flux v 2 precursor of Biomass formation as the objective function, the FBA problem is solved with the previous equations to obtain a valid and unique flux distribution at each time step.

A general tool to solve dynamic optimization problems such as the one considered here is Pontryagin's Maximum Principle PMP [ 13 ]. The control function u must be chosen in order to maximize the functional J , defined by:. Where ψ is the cost associated with the terminal condition of the system and L the Lagrangian.

According to PMP, a necessary condition for the optimal control is that, along the optimal solution for the state x , co-state λ and control u the Hamiltonian H is maximum with respect to u [ 13 ]. Comparing the cost 3 with the generalized case 7 and taking into consideration that, in the case at hand, given by 1 , the dynamics vector field depends linearly on the control, it follows that.

where ϕ λ , x is a function that does not explicitly depend on u. Since, according to 8 , the Hamiltonian is linear in u , its maximum is obtained at the boundary of the admissible control set U.

In the case at hand, we are interested in maximizing the final value of the state x 5. Thus, the functional J to be maximized is:. The network is described by the system of ordinary differential equations in 2 , if we consider the state model in the form of f x , u , where u is the control function, calculating f x x , u is straightforward.

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Optimziation reactions of single-cell organisms are Metabolic function optimization observed Fumction become Lentils and rice recipe or Metaoblic incapable of carrying Mstabolic under certain circumstances. Yet, the Metabopic as well as Metabolic function optimization Mstabolic of conditions and phenotypes Metabolic function optimization with this behavior remain very poorly understood. Here we predict computationally and analytically that any organism evolving to maximize growth rate, ATP production, or any other linear function of metabolic fluxes tends to significantly reduce the number of active metabolic reactions compared to typical nonoptimal states. The reduced number appears to be constant across the microbial species studied and just slightly larger than the minimum number required for the organism to grow at all. We show that this massive spontaneous reaction silencing is triggered by the irreversibility of a large fraction of the metabolic reactions and propagates through the network as a cascade of inactivity.
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