An Inactivation Switch Enables Rhythms in a Neurospora Clock Model

Abstract

Autonomous endogenous time-keeping is ubiquitous across many living organisms, known as the circadian clock when it has a period of about 24 h. Interestingly, the fundamental design principle with a network of interconnected negative and positive feedback loops is conserved through evolution, although the molecular components differ. Filamentous fungus Neurospora crassa is a well-established chrono-genetics model organism to investigate the underlying mechanisms. The core negative feedback loop of the clock of Neurospora is composed of the transcription activator White Collar Complex (WCC) (heterodimer of WC1 and WC2) and the inhibitory element called FFC complex, which is made of FRQ (Frequency protein), FRH (Frequency interacting RNA Helicase) and CK1a (Casein kinase 1a). While exploring their temporal dynamics, we investigate how limit cycle oscillations arise and how molecular switches support self-sustained rhythms. We develop a mathematical model of 10 variables with 26 parameters to understand the interactions and feedback among WC1 and FFC elements in nuclear and cytoplasmic compartments. We performed control and bifurcation analysis to show that our novel model produces robust oscillations with a wild-type period of 22.5 h. Our model reveals a switch between WC1-induced transcription and FFC-assisted inactivation of WC1. Using the new model, we also study the possible mechanisms of glucose compensation. A fairly simple model with just three nonlinearities helps to elucidate clock dynamics, revealing a mechanism of rhythms' production. The model can further be utilized to study entrainment and temperature compensation.

Keywords: Neurospora crassa; circadian clock; glucose compensation; mathematical modeling; molecular switch.

Figure A1

Kinetic parameters: A list of rate constants and their associated biological processes.

Figure A2

Co-existence of attractors: Bifurcation analysis of three parameters shows the co-existence of the limit cycle and steady state while the parameter is slowly varied. The values for increasing parameters are marked by orange circles. Decreasing parameters correspond to blue stars. Red lines mark jumps between steady states (low frq values) to limit cycle oscillations. The associated periods are given in the lower graphs. Default parameter values are marked by arrows.

Figure A3

Basins of attractors: Coexisting attractors shown in Figure A2 imply distinct basins of attraction (here, a3 = 3). Small initial values of frq and wc1 (red area) approach the steady state, whereas initial conditions in green approach the limit cycle. The initial conditions of the other variables were zero.

Figure A4

Essential and non-essential nonlinearities in the model: We linearized all three nonlinear terms and found that the Hill function of frq production (Term1) was essential, whereas the positive feedback nonlinearity (Term2) and the bilinear dimerization (Term3) were not essential since linearizations kept the rhythmicity.

Figure A5

Mutant simulations: Several experimentally-observed mutants were also reproduced by our model. The corresponding modified parameters are given in Table 1.

Figure A6

Sensitivity analysis after clamping the positive feedback: All parameters were changed by ±10 percent, and the resulting periods were calculated. Positive and negative period control coefficients indicate period lengthening and shortening, respectively (compare Figure 6).

Figure A7

Comparisons with the Hong model [15]: (A) There are no double peaks of the FRQnWC1n complex and a less pronounced switch (compare Figure 3). (B) The Hopf bifurcation is shifted to higher Hill coefficients (compare Figure 4).

Figure 1

Core network of Neurospora’s clock: The delayed negative feedback via FRQ is controlled by complex formations and phosphorylations.

Figure 2

Neurospora circadian clock model: The wiring diagram of the model shows compartmentalization into nucleus and cytoplasm, turnover of frq and wc1 colored in blue, complex formations, and nuclear translocation. The core components of the inactivation switch are marked by green, red, and yellow throughout the paper (A). Ordinary differential equations with 10 variables and 26 parameters (B).

Figure 3

Simulated time series: There are sinusoidal and spike-like waveforms, harmonics, and a temporal switch (see the text).

Figure 4

Bifurcation diagrams: The upper graphs show the maxima and minima of oscillations for varying parameters n (Hill coefficient) and a02 (WC1c overexpression). The lower graphs depict the corresponding periods. It turns out that oscillations persist for Hill coefficient n between 0.5 and 2.5, whereas overexpression of WC1 (a02) can terminate rhythms. Default parameter values are marked by bold numbers and arrows.

Figure 5

Inactivation switch in the Neurospora clock model: The FFCn levels (red line) are most of the time lower than the WC1n levels (green line), and only a small fraction of WC1n is bound to FFCn (yellow line) (see A). Below, we distinguish on and off phases. (B) points to an active WC1 (the thick line indicates strong binding to frq promoter) with a small degradation rate a18 (thin line). (C) illustrates the off state with FFC-assisted inactivation of WC1. Here, we have a strong binding of FFC to WC1 (a19 > a20) and fast degradation (a21 > a18).

Figure 6

Sensitivity analysis: All parameters were changed by ±10 percent, and the resulting periods were calculated. Positive and negative period control coefficients indicate period lengthening and shortening, respectively.