Flash-induced oxygen evolution in photosynthesis: simple solution for the extended S-state model that includes misses, double-hits, inactivation, and backward-transitions

Abstract

Flash-induced oxygen evolution in higher plants, algae, and cyanobacteria exhibits damped period-four oscillations. To explain such oscillations, Kok suggested a simple phenomenological S-state model, in which damping is due to empirical misses and double-hits. Here we developed an analytical solution for the extended Kok model that includes misses, double-hits, inactivation, and backward-transitions. The solution of the classic Kok model (with misses and double-hits only) can be obtained as a particular case of this solution. Simple equations describing the flash-number dependence of individual S-states and oxygen evolution in both cases are almost identical and, therefore, the classic Kok model does not have a significant advantage in its simplicity over the extended version considered in this article. Developed equations significantly simplify the fitting of experimental data via standard nonlinear regression analysis and make unnecessary the use of many previously developed methods for finding parameters of the model. The extended Kok model considered here can provide additional insight into the effect of dark relaxation between flashes and inactivation.

FIGURE 1

Flash-induced O₂ evolution in spinach chloroplasts published by Forbush et al. (1971). Original data were independently digitized twice, averaged, and normalized.

FIGURE 2

The Kok model of O₂ evolution, consisting of four stable states (S₀, …, S₃) and one transient state (S₄). The dashed arrows indicate dark relaxation of S-states.

FIGURE 3

Flash-induced transitions for S-states in the classic (A) and extended (B) Kok models of O₂ evolution. For simplicity, only transitions for S₂ are shown. The classic Kok model (A) has four stable states, whereas the extended Kok model (B) has five stable states, one of which (S_in) is an inactive state of OEC.

FIGURE 4

Analysis of the data of Forbush et al. (1971) using the classic Kok model (A–C) and the extended Kok model (D–F). Ten sequential oxygen evolution yields were used for the estimation of parameters of the model. Parameters determined from oxygen yields induced by flashes 1–10 are shown for the initial flash-number 1; parameters determined from flashes 2–11 are shown for the initial flash-number 2; etc. (A) Misses and double-hits; (B) initial conditions for S-states; (C) errors; (D) misses and double-hits; (E) initial conditions for S-states; and (F) errors. Calculations in A–C were made using Eqs. 8 and 9, whereas calculations in D–F were made using Eqs. 7 and 9. All calculations were made using the nonlinear least-squares curve fitter provided in Origin 6.1 software (OriginLabs, Northampton, MA). No boundary was assumed for the inactivation parameter during the fitting. All other parameters and initial conditions were assumed to be non-negative. Errors in C and F are calculated using the expression [Σ_i(y_i −f(n_i))²]/(N–p), where y_i and f(x_i) are the measured and the calculated oxygen yields induced by the flash n_i, respectively; N is the number of data points used in the fit (here N = 10); and p is the number of free parameters in the fit (p = 5 in the case of the classic Kok model and p = 7 in the case of the extended Kok model).