A model for the circadian rhythm of cyanobacteria that maintains oscillation without gene expression

Abstract

An intriguing property of the cyanobacterial circadian clock is that endogenous rhythm persists when protein abundances are kept constant either in the presence of translation and transcription inhibitors or in the constant dark condition. Here we propose a regulatory mechanism of KaiC phosphorylation for the generation of circadian oscillations in cyanobacteria. In the model, clock proteins KaiA and KaiB are assumed to have multiple states, regulating the KaiC phosphorylation process. The model can explain 1), the sustained oscillation of gene expression and protein abundance when the expression of the kaiBC gene is regulated by KaiC protein, and 2), the sustained oscillation of phosphorylated KaiC when transcription and translation processes are inhibited and total protein abundance is fixed. Results of this work suggest that KaiA and KaiB strengthen the nonlinearity of KaiC phosphorylation, thereby promoting the circadian rhythm in cyanobacteria.

FIGURE 1

Regulatory mechanisms of KaiC phosphorylation by KaiB. Solid arrows indicate regulations or transitions, which are experimentally suggested. Broken arrows indicate ones, which we assume in this work. (A) KaiB is assumed to have multiple states in which only activated KaiB (KaiB*) enhances the dephosphorylation of KaiC. In panels B–D, three alternative regulations are also considered.

FIGURE 2

Bistability of KaiC phosphorylation. Abundance of phosphorylated KaiC protein x at equilibrium is plotted as a function of the total abundance of total KaiC, C. (A) Regulation of KaiC is described by Eq. 1 in the text. These diagrams have been obtained with XPPAUT (http://www.math.pitt.edu/∼bard/xpp/xpp.html). Regulation of KaiC is described by Eqs. 1a and 5 in panel B, by Eqs. 1a and 6 in panel C, and by Eqs. 1a and 7 in panel D. Parameter values in panel A are as follows: B₀ = 200, C₀ = 2, p = 6.3, a = 0.207, b = 0.063, f = 4, g = 25.2, k₁ = 630, q = 1.8, n = 2. Alternative interactions between KaiC and KaiB are also shown (B–D). Parameter values are a = 1.5, n = 8 (B), a = 1.5, n = 100 (C), and a = 1.5, n = 4 (D), and other parameter values except a and n are the same as in panel A.

FIGURE 3

Gene-protein networks of circadian rhythms in cyanobacteria. (A) Negative feedback regulation of kaiBC expression by KaiC. (B) Regulations of KaiA, KaiB, and KaiC in the DD condition where transcription-translation feedback is inhibited. (C) Negative feedback regulation of kaiBC expression incorporating the dynamics of KaiA.

FIGURE 4

Relaxation oscillations in the cyanobacterial circadian clock. (A) Abundance of phosphorylated KaiC oscillates under the regulation of total KaiB and KaiC (see Eq. 3). The timescale of transcription and translation is varied by changing ɛ₁. Numbers by the graphs are for ɛ₁. When ɛ₁ is small (i.e., ɛ₁ = 0.001), the amplitude of the cycle is large with a large period. When ɛ₁ is very large (i.e., ɛ₁ = 1), abundance of phosphorylated KaiC converges to steady state. This diagram has been obtained with XPPAUT. Parameters are as follows: λ = 15, h₁ = 1, μ = 6.3, and m = 4. Other parameter are the same as in Fig. 2 A. (B) Oscillations of the abundance of phosphorylated KaiC (indicated by the smooth curve) and activated KaiB (indicated by the dotted curve) for ɛ₁ = 0.07.

FIGURE 5

Persistence of KaiC phosphorylation rhythm in LL and DD. The dotted curve indicates abundance of KaiC. Abundance of KaiC oscillates in LL conditions in which the period of the oscillation is ∼24 h. Transcription, translation, and degradation of proteins are suppressed in DD conditions. Therefore abundance of KaiC is almost constant in DD. However the abundance of phosphorylated KaiC shown by the solid curve still oscillates in DD. Moreover the period of the oscillation of phosphorylated KaiC is similar in DD and in LL. Parameters are as follows: p = 2.5, b = 0.025, g = 10, k₁ = 250, ɛ₁ = 0.02, λ = 5.95, μ = 2.5, ɛ₂ = 0.02, l = 2.5, w = 0.551, k₂ = 2.5, and h₂ = 1. The other parameters are the same as in Fig. 4 B.