Coupled Simulation of Bioreactor and Metabolic Network Dynamics By Ordinary Differential Equations with Embedded Optimization

Coupled simulation of bioreactor and metabolic network dynamics by ordinary differential equations with embedded optimization

Xiao Zhao, Stephan Noack, Eric von Lieres, Wolfgang Wiechert

IBG-1: Biotechnology, Forschungszentrum Jülich, Germany

We present a novel modelling framework, namely ordinary-differential equations with embedded optimization (ODE-O), for combined simulation of microbial conversion processes in bioreactor environments. The framework comprises two interrelated levels. The upper level is a classical dynamic system in the form of an ODE. Mass balances are defined for the components in the liquid of the bioreactor, i.e., biomass, substrates and products. The lower level determines fluxes among intracellular metabolite pools and through cell boundaries by solving a nonlinear optimization problem. Both levels are coupled by uptake and secretion rates of substrates and products. The resulting modeling and simulation framework allows to systematically study the combined dynamics of bioreactors and cultivated cells.

The general mathematical form of an ODE-O comprises a dynamic (upper level) model, eq. (1a) and a (lower level) optimization, eq. (1b):

dx=f(x,v), x(0)= x0	(1a)
v=arg minv g(x,v) st.0 = h(x,v)    0 ≥ l(x,v)	(1b)

Here, x in R^m and v in Rⁿ, f, h, l are smooth vector-valued functions, g is a smooth real-valued function, x₀ in R^m denotes a vector of initial conditions, x and v are the state and algebraic system variables, respectively.

The solution of ODE-O models with nonlinear inner optimization is hardly addressed in the literature. System (1) can be solved by transforming the inner optimization problem, eq. (1b), into a set of algebraic equations, using the first-order necessary optimality conditions (KKT condition), resulting in a (quasi) DAE model (Landry et al. 2009). This approach has the advantage that off-the-shelf DAE simulators can be applied without solving the inner optimization problem directly. Customary DAE simulators require that the generated DAE system has index 1, which is guaranteed by the following assumptions:

(1)   Local uniqueness of the minimizer of the inner optimization problem, eq. (1b).

(2)   Linear Independent Constraint Qualification (LICQ) condition, i.e. the gradients of all active constraints of the inner optimization problem, eq. (1b), are independent. 

However, bioreactor models with embedded metabolic networks do not necessary fulfil the LICQ condition. Consequently, the classical KKT based reformulation cannot be integrated by customary DAE solvers. Hence, we propose an alternative method that is based on Extreme Flux Analysis (Klamt et al., 2003) and transforms the inner optimization problem into an equivalent form, for which the LICQ condition is fulfilled under mild conditions. Our novel framework has been applied to model an amino acid production process with Corynebacterium glutamicum. The resulting simulation data will be discussed in detail.  

References

Landry, C., Caboussat, A. and Hairer, E., 2009. Solving optimization-constrained differential equations with discontinuity points, with application to atmospheric chemistry, SIAM Journal on Scientific Computing 31 (5), 3806-3826.

Klamt, S. and Stelling, J., 2003. Two approaches for metabolic pathway analysis. Trends in Biotechnology 21 (2), 64-69.