State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, PR China

Department of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, NY 11790, USA

Graduate School of the Chinese Academy of Sciences, Beijing, 100039, PR China

Abstract

Finding the global probabilistic nature of a non-equilibrium circadian clock is essential for addressing important issues of robustness and function. We have uncovered the underlying potential energy landscape of a simple cyanobacteria biochemical network, and the corresponding flux which is the driving force for the oscillation. We found that the underlying potential landscape for the oscillation in the presence of small statistical fluctuations is like an explicit ring valley or doughnut shape in the three dimensional protein concentration space. We found that the barrier height separating the oscillation ring and other area is a quantitative measure of the oscillation robustness and decreases when the fluctuations increase. We also found that the entropy production rate characterizing the dissipation or heat loss decreases as the fluctuations decrease. In addition, we found that, as the fluctuations increase, the period and the amplitude of the oscillations is more dispersed, and the phase coherence decreases. We also found that the properties from exploring the effects of the inherent chemical rate parameters on the robustness. Our approach is quite general and can be applied to other oscillatory cellular network.

**PACS Codes:** 87.18.-h, 87.18.Vf, 87.18.Yt

1.Introduction

Circadian rhythms are an intracellular timing mechanism, widespread in living organisms, with a period of about 24 h, which fits the day/night alterations of the Earth adapting to the fluctuating environment. In Neurospora, Arabidopsos, Drosophila, and mammals, transcription-translation-derived oscillations originating from negative feed back regulation of clock genes have been modeled at the molecular level. The study of the oscillation behavior in an integrated and coherent way is crucial in modern systems biology for understanding how these rhythms function biologically. The underlying natures of the rhythmic behavior have been explored by many experimental and theoretical methods

We decided to explore an established basic model based on known biological and biochemical features of a circadian clock which has negative regulation _{N }in a reversible manner. Finally, _{N }represses the transcription of the gene. Effectively, therefore, the network is like a self repression with time delay, in which oscillation behavior is expected.

Model

**Model**. Model for the molecular mechanism of circadian rhythms in Drosophila.

It is important to demonstrate the robustness and stability issues of the circadian system and associated oscillation patterns. There are many possible states in the systems, and it is difficult to explore all of them and the associated global nature of the network

Therefore, instead of the averaged deterministic network of chemical rate equations, we developed a probabilistic description to model the corresponding cellular process taking into accounts of the intrinsic and external fluctuations. This can be realized by constructing a master equation for the intrinsic fluctuations or the diffusion equation for external fluctuations of the time dependent evolution probability rather than the average concentration for the corresponding deterministic chemical reaction network equations

By solving the Fokker-Plank diffusion equation, we can obtain the probability distribution in protein concentrations evolving in time. We can also uncover the long-time steady-state probability of this chemical reaction network in analogy to the equilibrium system, where the global thermodynamic properties can be explored using the inherent interaction potentials. We will study the global stability by exploring the underlying potential landscape for the above-mentioned non-equilibrium protein network. The generalized potential energy can be shown to be closely associated with the steady state probability of the non-equilibrium network in general and has been applied to a few systems

The adaptive landscape idea was first introduced into biology by S. Wright, Delbruck and Waddington

A deterministic mathematical model of this protein clock constrained by experimental data has been proposed recently

where _{N }is the concentration of the nuclear forms of the clock protein. The parameter _{s }represents maximum rate of transcription, and _{m }is the maximum rate of transfer into the cytosol, with the Michaelis constant _{m}. _{s }is the rate of protein synthesis, and _{d }is the maximum rate of protein degradation, with Michaelis constant _{d}. _{1 }and _{2 }are the first-order rate characterizing the transport of the protein into and out of the nucleus. The negative autoregulatory feedback is the origin of the oscillations.

As mentioned, the statistical fluctuations may be significant from both internal and external sources _{N}:

where **x **= (_{N}). We can rewrite the diffusion equation as **x**,

or in the component notation, _{1}(_{N}, _{1}(_{N})_{2}(_{N}, _{2}(_{N})_{3}(_{N}, _{3}(_{N})**x**, _{ss }stands for steady state probability. Although **F **in general can not be represented as a potential gradient, the driving force for the dynamics can be decomposed to two terms for non-equilibrium network systems: one is associated with the gradient of a potential closely linked to the steady state probability and the other is associated with a divergent free field. The divergent free field has no sources or sinks to start or end the force lines and therefore is recurrent or rotational in nature

Once we solve for the steady state probability from the probabilistic diffusion equation, we can study the underlying properties of the potential (or potential landscape) by the relation: **x**) = -ln **x**)

2. Results and discussion

The parameter values are _{s }= 0.5 ^{-1}, _{m }= 0.3 ^{-1}, _{d }= 1.5 ^{-1}, _{s }= 2.0 ^{-1}, _{1 }= 0.2 ^{-1}, _{2 }= 0.2 ^{-1}, _{m }= 0.2 _{d }= 0.1

We solve the Fokker-plank equation using both the reflecting boundary condition J = 0 and the absorbing boundary condition. The results are similar; we choose the reflecting boundary condition in this paper. With certain initial conditions(both homogeneous and inhomogeneous), we obtain the steady probability distribution solution _{ss }using the finite difference method at the long time limit. Then, we can use **x**) = -ln _{ss}(**x**) to get the generalized non-equilibrium potential function (landscape) of the circadian clock.

In order to see the results clearly, we can integrate the three dimensional probability _{N},

The integrated results are shown in Fig.

Integrated 2 dimensional potential landscape

**Integrated 2 dimensional potential landscape**. The integrated two dimensional effective landscapes for the three dimensional system.

Fig.

3 dimensional potential landscape

**3 dimensional potential landscape**. The three dimensional potential landscape and flux for

We can clearly see the probability distribution is not distributed evenly along the limit cycle. In order to know the nature of attractive nature of the limit cycle, we have to observe the dynamics of the network. The deterministic oscillation for the three variables _{N }over a period are shown in Fig. _{N }over the period are shown in Fig.

Speed

**Speed**. A: the deterministic oscillation for the three variables _{N }over a period. B: the forces of _{N }over the period. C: the speed along the cycle with time. The 'star' time parameters are as follows:_{1 }= 3.8, _{2 }= 8.5, _{3 }= 15.5, _{4 }= 21.2. D:The speed along a limit cycle: the 'star' time parameters are the same as Fig. 4C.

The divergence of the flux is equal to zero at steady state. In an equilibrium system, the flux **J **= 0 (detailed balance). But in a non-equilibrium system, the flux is a curl field (

So the flux force is the driving force for the oscillation. The potential landscape attracts the system to the closed ring and the flux force keeps the probability flow along the ring, providing the driving force for oscillation. We can see that the flux force plays a more important role along the closed ring than outside the ring because of large ∇

We can explore the global stability and robustness of the circadian clock when we obtain the potential landscape. The barrier height represents the system escaping from the oscillation attractor. Fig. _{fix }minus _{max}, and _{fix }minus _{min}, where _{fix }is the potential local maximum inside the limit cycle; _{max }is the potential maximum along the limit cycle; and _{min }is the potential minimum along the limit cycle. We can see the barrier height becomes larger when the fluctuations decrease. It is harder for the system to go from the doughnut of attraction to outside when fluctuations are small. This means the doughnut shape of the landscape is robust, and a stable oscillation is essentially guaranteed for small fluctuations. It also implies that the barrier height can be used as a quantitative measure of the stability and robustness of the network oscillations.

Barrier

**Barrier**. The barrier height _{fix }- _{min }and _{fix }- _{max }versus diffusion coefficient

In a non-equilibrium open system, there are constant exchanges of energy and information with the outside environment. This results in the dissipation of energy, which gives a global physical characterization of the non-equilibrium system. The circadian clock is a non-equilibrium open system. In the non-equilibrium steady state, the system still dissipates energy and entropy which can be determined using the landscape and the flux globally, where the entropy production rate is equal to heat dissipation. In the steady state, the dissipation of energy is closely associated with the entropy production rate. The entropy formula for the system is given as

_{B }∫ **x**, **x**,

By differentiating the above function, the increase of the entropy at constant temperature

where _{p }= -∫(_{B}**F**)·**J **_{d }= ∫ **F**·**J **_{p }is equal to the heat dissipation _{d}. In Fig.

EPR

**EPR**. A:The diffusion coefficient _{fix }- _{min }and _{fix }- _{max }versus the entropy production rate.

The robustness of the oscillation with respect to the diffusion coefficient _{1}(_{1 }+ _{2}(_{2 }+ _{3}(_{3 }is shown in Fig. _{1 }= (0, 1), _{2 }= (-_{3 }= (-_{1}(_{2}(_{3}(^{-1}. The oscillation goes in the positive orientation (counterclockwise), so

Definition of phase coherence

**Definition of phase coherence**.

where

Coherence

**Coherence**. A: The coherence versus the diffusion coefficient

We can also use stochastic simulations for various values of

Period

**Period**. A: The period distribution for _{Period }and the entropy production rate. D: The standard deviation of period _{Period }versus the barrier height.

We also show the distributions of the amplitude for

Amplitude

**Amplitude**. A: The amplitude distribution with different

To explore the effects of the inherent chemical rate parameters on the robustness, we can try to find out which reactions are important, and further, which protein elements are crucial in maintaining the robustness. Fig. _{1 }and _{m }increase, the barrier height increases and the entropy production rate decreases, as the system becomes more stable and robust. We can also see that when the other four rate parameters (_{2}, _{s}, _{d}, _{s}) increase, the barrier height decreases and the entropy production rate increases as the system becomes less stable and robust.

Barrier height and entropy production versus chemical rate parameters

**Barrier height and entropy production versus chemical rate parameters**. A: Barrier changes with respect to changes of rate parameters (red: rate increase. green: rate decrease) _{1}, _{2}, _{s}, _{m}, _{d}, and _{s}. B. Barrier height and entropy production rate versus chemical rate parameters _{1}, _{2}, _{s}, _{m}, _{d}, and _{s}.

We can choose the rates _{s }and _{m }to further explore the period and amplitude using stochastic Brownian dynamics, since they represent the largest changes of the barrier height from the increasing the rate parameters. Fig. _{s }rates. Fig. _{s }increase. Fig. _{s }rates and Fig. _{s }increase. This implies that the fluctuations in period measured by the variance increase as the _{s }increases. Therefore the network becomes less stable and coherent due to the trend of larger fluctuations.

Amplitude and period distribution changes with respect to rate _{s}

**Amplitude and period distribution changes with respect to rate k**

Fig. _{m }rate. Fig. _{m }increase. Fig. _{m }rate and Fig. _{m }increase. This implies that the fluctuations in period and amplitude measured by the variance decrease as the _{m }increases. Therefore the network become more stable and coherent due to the trend of smaller fluctuations.

Amplitude and period distribution changes with respect to rate _{m}

**Amplitude and period distribution changes with respect to rate v _{m}**. A: Amplitude distribution for different chemical rates

2.1. Mathematical material

We can study the network of chemical reactions in fluctuating environments:

where **x **= {_{1}(_{2}(_{n}(**F**(**x**) = {_{1}(**x**), _{2}(**x**), ... _{n}(**x**)} is the chemical reaction rate flux vector involving the chemical reactions which are often non-linear in protein concentrations **x **(for example, enzymatic reactions). The equations **F**(**x**) describe the averaged dynamical evolution of the chemical reaction network (see details in the next subsection). As mentioned, in the cell, the fluctuations can be very significant from both internal and external sources

**x**, ^{τ }(**x'**, **x**,

Here _{i}(_{j}(_{ij}

We can explore the long time steady state properties and collect the statistics to obtain the steady state distribution function _{0}(**x**) for the state variable **x **(representing the protein concentrations of the protein network in this case). In the equilibrium systems where a potential U where the force is a gradient of it, _{0}(**x**) is exponentially related to potential energy function **x**). So we obtain the information of steady state probability from U. For the non-equilibrium system, we do not know the information of the potential a priori. But we can obtain the information of the steady state probability by solving the probabilistic evolution equation and taking the long time limit. In analogy with the equilibrium system, we can define the generalized potential U for the non-equilibrium case from the steady state probability

with the partition function **x **exp{-**x**)}. The rational for the definition of the non-equilibrium potential this way is given earlier in the main text due to the driving force (for the dynamics) decomposition as gradient of a potential and curl flux. From the steady-state distribution function, we can therefore identify U as the generalized potential function of the network system. In this way, we map out the potential landscape. Once we have the potential landscape, we can discuss the global stability of the protein cellular networks.

3. Conclusion

We have shown that we can explore the global features of the circadian rhythms model. Finding the potential landscape and associated flux is the key to addressing the robustness issue of the networks. We have uncovered the underlying potential landscape of a circadian clock. This is realized by explicitly constructing the probability of the states of the protein network by solving the corresponding probabilistic diffusion equation. The landscape of the oscillation has an irregular and inhomogeneous closed ring valley or doughnut-like shape. We also found that the flux along the cycle path is the driving force for coherent oscillation. The potential barrier height for escaping from the limit cycle attractor determines the robustness and stability of the network oscillations. We found as the diffusion coefficient becomes smaller, the potential barrier becomes greater, and furthermore the statistical fluctuations are effectively more severely suppressed. This leads to robustness of the biological limit cycle basin of the protein network.

We observe the global dissipation in terms of the entropy production of the whole system increases when the diffusion coefficient

The robustness, coherence, and dissipation of the circadian oscillations with respect to the changes with the rate parameters can be studied as well. And we found protein element k_{s }is crucial in maintaining the robustness in the network.

Acknowledgements

JW thank the supports from National Science Foundation Career Award and American Chemical Society Petroleum Fund. LX and EKW are supported by the National Natural Science Foundation of China (Grant no. 90713022 and 20735003).