Model of photosynthetic factory – PSF model

Here we characterize the multi time-scale three-state model of photosynthetic factory (PSF model) proposed by Eilers and Peeters [24, 25] and further developed by Wu and Merchuk [28, 29] and Celikovsky, Papacek and Rehak [26, 27, 30–32]. PSF model (Fig. 1), is used for derivation of the reaction term R(c_i), see the transport Eq. (1).

Scheme of Photosynthetic factory model (PSF). Four parameters (α, β, γ, δ) describe transitions among respective states (resting R, activated A, and B inhibited)

PSF model (7) incorporates the dynamics of three fundamental states propagating in different orders of time scales: (i) cell growth (including the shear stress effect), (ii) photoinhibition, and (iii) photosynthetic light and dark reactions. The state vector of the PSF model is three dimensional, y=(y_R,y_A,y_B)^⊤, where y_R indicates the probability that PSF is in the resting state R, y_A the probability that PSF is in the activated state A, and y_B the probability that PSF is in the inhibited state B.

The values of PSF model parameters, i.e., α, β, γ, δ, see (7), and κ, see (6), are taken from original study [28], where Wu and Merchuk identified these values of PSF model parameters for the microalga Porphyridium sp.: α = 1.935 × 10⁻³ μE⁻¹ m², β = 5.785 × 10⁻⁷ μE⁻¹ m², γ = 1.460 × 10⁻¹ s ⁻¹, δ = 4.796 × 10 ⁻⁴ s ⁻¹, κ = 3.647 10 ⁻³ and Me = 0.059 h ⁻¹. For details regarding the experimental setup, for parameter estimation and the identifiability study, see [31, 32].

In order to make the PSF model parameter estimation more robust, the following re-parametrization was introduced and the singular perturbation method was used in [31]: q1:=γδαβ,q2:=αβγδ(α+β)2,q3:=κγαδβγ,q4:=αq1,q5:=β/α. Consequently, the PSF model acquires the following form:

For the given input variable, i.e., the irradiance u(t), the ODE system (7) can be solved either by numerical methods or by asymptotic methods. However, the irradiance level of 250 μE m ⁻² s ⁻¹ (the value of the parameter q₁ which maximizes the steady-state growth [31]), provides two negative eigenvalues of the (7) system matrix: λ₁ = -0.63, λ₂ = -0.59 10 ⁻³. The ODE system (7) is stiff due to value 10³ of the ratio λ1λ2. This fact points to the existence of two processes: (i) photosynthetic reactions, and (ii) photoinhibition, widely separated from the point of view of their characteristic times. Further, without significant loss of accuracy, we use the so-called fast reduction [27], i.e., the behavior of the system (7) is characterized by the only one following ODE

and the “slow” variable y_B can be regarded as a constant depending on the averaged value u=u_av. According to our works [27, 33] it holds

Both the non-reduced (7) and the reduced order PSF model Eq. (11) have been incorporated as a User-Defined Function (UDF) in ANSYS Fluent, see the following section.