How to deal with the multi-scale dynamical systems: A case study – Model for Photosynthesis, Photoinhibition and Photoacclimation

Š. Papáček¹, S. Čelikovský⁴, D. Štys^1,2,3

¹Institute of Physical Biology, University of South Bohemia,

² Institute of Systems Biology and Ecology CAS, Zámek 136, 373 33 Nové Hrady, Czech Republic

³Institute of Microbiology CAS, 379 81 Třeboň, Czech Republic.

⁴UTIA AV ČR (Institute of Information Theory and Automation, Academy of Sciences of the

Czech Republic), Prague, Czech Republic.

papacek@greentech.cz

The multi-scale modelling and computation became essential in solution process of multi-scale systems frequently arising in chemistry and biology. The systems with multiple time-scales are characterised by the wide separation between the O(e) time-scale, and the O(1), or even O(e^-1). In this paper we describe three different ways to solve the problem of microalgal photosynthesis under periodic intermittent light. The lumped parameter model for photosynthesis, photoinhibition and photoacclimation in microalgae represents the so-called stiff system, i.e. the dynamical system with multiple time-scales. The separation of time-scales in the system is modeled by a small parameter e. From a mathematical point of view, the problem in hand is a system of two ordinary non-linear differential equations (ODE), and we have assumed that the phase-space can be decomposed into the slow phase, x₃, and the fast one, x₂. Ordinarily, the ODE system is not amenable to analytical treatments and requires an efficient computational tool for its solution. However, in our case, i.e. for the piecewise constant input (light intensity u), is the analytical solution reachable because we deal with the linear ODEs in both subintervals of light-dark cycles period: h_a - light-off interval, and h_b - light-on interval. Consequently, we are able to evaluate the average value of the fast phase x₂ (which is reflecting the rate of photosynthesis in microalgal culture) depending on the period of light-dark cycles h = h_a + h_b. Next, we can analytically compute the solution of the same ODE system applying two common approaches for solving the stiff systems: (i) neglecting the fast system dynamics, and (ii) neglecting the slow dynamics. Once having the exact analytical solution, we can observe the limits of validity of both approximated solutions. The presented simulation results based on published data demonstrate the perfect concordance between the theoretical assumptions and the numerical results and could serve as the motivation example for applying the multi-scale method, and at the same time, to eliminate some frequent errors in treatment of stiff systems.