There is “fluctuation dissipation theorem”. Porf Hilfinger reminded me to be cautious of this term in physics context. To have a taste of it, you can check Linear Response - Biophysics.

The content of this post is closely linked to the research conducted by A. Hilfinger, J. Paulsson, and J. Elf, etc. As a mathematical rookie, I will introduce the models and mathematics in the most accessible way. You need a second-year undergrad level of mathematics to read this.

At the starting point, 3, 5 and 10 in the references are three papers can be read in details.

Noise in Gene Regulatory Networks

In 2002, Elowitz et al. provided experimental evidence that gene regulatory networks inherently exhibit noise 1. How to analyze the noise then? In Molecular Biology (a course I took as an undergrad, and I will write something in Chinses, but haven’t started yet…), we’ve already learned how complicated the mechanisms of transcription and translation could be. However, physicists are people having good intuition of extracting from facts and making up models to describe the world. They have tried to use Markov processes to describe the network with intrinsic noise (discussed in van Kampen’s IV.1 2, also quoted in (En) Stochastic Processes). We describe the dynamics of the number (integers) of molecules (mRNAs, proteins…) as discrete birth and death jump between the grids of the whole phase space.

More about the model construction, see Appendix of 3 and the main text of 4, also quoted in (En) Stochastic Processes.

One-Variable Case

Always start from the simplest! Let’s consider a molecule \(X\)’s probability (per unit time) of production and degradation are \(f(x)\) and \(g(x)\) (also refer to transition rates). As a Markovian process, we take the transition rates merely depends on where the current state is. We write it as

which the probability distribution on different grids is \(P(x,t)\). The time evolution of the probability distribution \(\frac{dP(x,t)}{dt}\) is described by a master equation, which can be interpreted as “plus the probability in and minus the probability out that state”. From \(\frac{dP(x,t)}{dt}\), using the “index shift” trick, we can get the evolution of statistical quantities of \(X\) (average over the ensemble). (This is from Hilfinger’s lecture note of PHY2707, and I don’t know whether there is copyright. If there is, I will sincerely すみません…)

Special case: constant birth and first-order degradation

Now let’s look at a intuitive and simple case where the molecule has a constant production rate and a first-order degradation rate. The steady state (also detailed balance in this case) of the probability distribution on grids forms a Poisson distribution. (from Hilfinger’s lecture note…)

Multi-variable Case

Gene regulatory networks consist of many interacting molecule species. We consider a biochemical system with \(n\) molecular species and \(m\) elementary reactions.

For this network, also using the “index shift” trick to calculate mean, variances and covariances with master equation. The detailed steps can be found in 5, equation (1)-(3). I don’t bother to type the results here.

In these jump models, the diffusion and drift coefficients are related, by means of a Lyapunov equation 6, to the steady-state covariance matrix 5. If all the transition rates are linear functions, the Lyapunov equation is an exact result from the master equation and no Gaussian approximation is needed. If the transition rates are nonlinear, van Kampen’s \(\Omega\) expansion (leads to linear noise approximation) is needed to derive the equation.

We follow the content in 5. Define the vector of transition rate as \(f(x):=\sum_ir_iW_i(x)\).

Linear Case

We can write the transition rate as \(f(x(t))=Ax(t)+f_0\), where \(A\) is an n*n real constant matrix (drift) and \(f_0\) is an n-dimensional constant vector.

Define the covariance matrix \(\Sigma(t):=\langle x(t)x^T(t)\rangle-\langle x(t)\rangle\langle x^T(t)\rangle\), autocorrelation \(R_{xx}(t_1,t_2):=\langle x(t_1)x^T(t_2)\rangle-\langle x(t_1)\rangle\langle x^T(t_2)\rangle\). Following 4, there are some important exact results.

\(\frac{d\Sigma(t)}{dt}=A\Sigma(t)+\Sigma A(t)^T+D(t)\). (5 in text, where \(D(t):=\sum_ir_iW_i(\langle x(t)\rangle)r_i(t)\))

\(\frac{\partial R_{xx}(t_1,t_2)}{\partial t_1}=AR_{xx}(t_1,t_2)\). (7 in text)

For steady state,

\(AR_{xx}(0)+R_{xx}(0)A^T+D=0\). (9 in text) This is the Lyapunov equation, and often called the fluctuation dissipation theorem, a balance between the tendency of particles to diffuse and mechanisms such as negative feedback that bring them back to equilibrium (Remark 3 in text).

\(\frac{\partial R_{xx}(\tau)}{\partial\tau}=AR_{xx}(\tau)\). (10 in text) And how to solve this systems of ODEs? Use matrix exponential, which involves diagonalizing the matrix, see Linear Algebra (线性代数 - Matrix Exponential).

The Classical Model of mRNA and Protein

This is an example of linear case with the classical mRNA-protein model 3. You can regard \(X_1\) as the mRNA and \(X_2\) as the protein. (In many case, the protein degradation rate is very small compared with cell growth 7, and the dilution effects are often described as first-order degradation.) You will come across this model from time to time when you are reading Paulsson group related things… For example, model [4] in 8, and equation [7] is the exact result of this linear rates model. See below how to derive.

The Classical Model with Additional Complex Formation and Disassociation

In 9, the role of miRNA-mRNA complex in regulating noise in protein level was considered. In the following figure I showed the matrix \(D\) of the reaction involves more than one species.

The “Normalized” Form

Well, I base the above derivation on 5. In the well-known paper 9, where the author set up a “standard procedure”, they are in a normalized form (example papers using the “normalized” form: 9, 10, 11). I show the procedure to normalize them. Note that “Little’s Law” is applied here.

From Langevin Equation

For the linear case, the Lyapunov equation, or the fluctuation dissipation theorem, can also be derived from the Langevin equation \(\frac{d \vec{a}}{dt}=H\vec{a}+\vec{\tilde{f}}\), where \(\vec{\tilde f}(t)\) is a white noise vector. The steps can be found on chapter1.8 from 12. You can also find my note Network_Langevin_Lyapunov here.

The relationship between master equation, Fokker-Planck approximation and Langevin approach will be discussed in the following part.

Nonlinear Case

In the nonlinear case, we cannot simply drag the drift matrix \(A\) out as \(f(x(t))-\langle f(x(t))\rangle=A(x(t)-\langle x(t)\rangle)\) (above equation 5 in 5), and get the nice Lyapunov equation. To quantify the variances and covariances (second moment properties), we need to do some approximation…

van Kampen’s \(\Omega\) expansion and Linear Noise Approximation

The \(\Omega\)-expansion means that the Master equation is Taylor expanded near macroscopic system trajectories or stationary solutions in powers of \(1/\sqrt{\Omega}\) , where \(\Omega\) is the system volume. When the Master equation is approximated near a macroscopically stable stationary solution, terms of first order in \(1/\sqrt{\Omega}\) give the macroscopic rate equations, and terms of second order in \(1/\sqrt{\Omega}\) give the Linear Noise Approximation (LNA). The rationale behind this approach is that for constant average concentrations, relative fluctuations will tend to decrease with the inverse of the square root of the volume 13.

P.S. In most papers, researchers look at the stationary properties. A cyclostationary case (not stationary) case where Linear Noise Approximation is applied near macroscopic system trajectories can be found in 14.

You can find a formal explanation, check chapter X of 2. If you are looking for a shorter and closely related to gene regulatory network text, check **the supplementary material of 13 ** (after “Derivation of the LNA”, go to “Example 3: Coupled fluctuations in a two component systems” starting from page 6).

I would like to explain it as “changing the variables from number of species to the rescaled fluctuations of species”. We start from the nonlinear master equation and the corresponding joint probability distribution of species (equation A3 in supplementary materials of 13). After approximation, we end at the linear Fokker-Planck equation about the joint probability distributions of the fluctuations \(\frac{\partial\Pi(\xi,t)}{\partial t}=-\displaystyle\sum_{ik}A_{ik}\frac{\partial}{\partial \xi_i}\xi_k\Pi(\xi)+\frac{1}{2}\displaystyle\sum_{ik}[BB^T]_{ik}\frac{\partial^2\Pi(\xi)}{\partial\xi_i\partial\xi_k}\) (equation A9 in supplementary materials of 13).

P.S. An alternative perspective of LNA can be found in Section A.3 of the Supplemental Information in 15, which is \(R_i^+(x)\approx R_i^+(\langle x\rangle)+\nabla R_i^+(\langle x\rangle)\cdot(x-\langle x\rangle)\). Clearly, the physical meaning is expand near macroscopic system trajectories or stationary solutions. Where matrix \(\nabla R_i^+(\langle x\rangle)\) can be used as the rate matrix as in the linear case.

Solution of Linear Fokker-Planck Equation

The stationary solution is equation A11(in supplementary materials of 13), the multidimensional Normal/Gaussian distribution, and the detailed steps can be found in chapter 6.5 Ornstein-Uhlenbeck Process from 16. Generally speaking, Fourier transform will be applied to solve this multivariable PDE. First have a taste by my note Biophysics_Diffusion (diffusion equation from Biophysics on the simplest case and see how Fourier transform works).

Lyapunov Equation of Linear Fokker-Planck Equation

From \(\frac{\partial\Pi(\xi,t)}{\partial t}=-\displaystyle\sum_{ik}A_{ik}\frac{\partial}{\partial \xi_i}\xi_k\Pi(\xi)+\frac{1}{2}\displaystyle\sum_{ik}[BB^T]_{ik}\frac{\partial^2\Pi(\xi)}{\partial\xi_i\partial\xi_k}\), simply multiply by \(\xi_j\xi_l\) and integrate over the whole phase space, and we will get the Lyapunov equation \(\frac{\partial \Xi(\xi,t)}{\partial t}=A\Xi+\Xi A+BB^T\), where the covariance matrix \(\Xi=\langle \xi\xi^T\rangle\) is the covariance matrix of rescaled fluctuations. In stationary, \(0=A\Xi+\Xi A+BB^T\)

The hint is from VIII.6 2, and I am writing down the formal steps. If you have problems in these steps, I highly recommend you to take PHY2315, (En) Advanced Statistical Mechanics (my lecturer is Prof Dennis Dalidovich and he wrote excellent notes!). After that, you will have better sense of related math.

Example of Nonlinear Case

Check “Example 3: Coupled fluctuations in a two component systems” in supplementary materials of 13.

The method of using the Lyapunov equation to compute the variance and covariance of species is frequently employed in the research of A. Hilfinger, J. Paulsson, J. Elf, etc. You can read 9, the famous paper by J. Paulsson where the Lyapunov equation is presented intuitively (and my post derived Equation (S4) in a complete and rigorous way).

Master Equation, Fokker-Planck Approximation and Langevin Approach

We describe the stochastic process by Master equation \(\frac{\partial P(y,t)}{\partial t}=\int\{W(y\vert y^\prime P(y^\prime,t)-W(y^\prime \vert y) P(y,t)\}\), corresponding to a stationary joint statistical properties (mean, variances, covariances…). However, it is unusually impossible to solve the statistical properties directly from the master equation, unless the transition rates are linear.

To solve it approximately, Fokker-Planck equation is based on the assumption that the jumps are relatively very small. N. G. van Kampen’s \(\Omega\) expansion is a formal method when the system is large (jumps are relatively small). Intuitively, we get a solvable linear Fokker-Planck from the expansion. The Langevin approach is mathematically equivalent to Fokker-Planck equation. In the linear case, I derived the Lyapunov equation from Langevin approach. To read more, check chapter V. VII. and VIII. from 2.

Wrap up

I sincerely thank everyone in my group, especially B Kell and A. Hilfinger. You can regard this post as my first-year progress in theory, although in fact it should not take that long…. I feel like just crossing the critical point. I completed a phase transition of “being defeated by the equations from the beginning of papers” to “understand the basic math and have better sense of deriving statistical properties”.

References

[1] Elowitz, Michael B., Arnold J. Levine, Eric D. Siggia, and Peter S. Swain. “Stochastic Gene Expression in a Single Cell.” Science 297, no. 5584 (August 16, 2002): 1183–86. online.

[2] Van Kampen, Nicolaas Godfried. Stochastic processes in physics and chemistry. Vol. 1. Elsevier, 1992.

[3] Thattai, Mukund, and Alexander Van Oudenaarden. “Intrinsic noise in gene regulatory networks.” Proceedings of the National Academy of Sciences 98.15 (2001): 8614-8619. online

[4] Paulsson, Johan. “Models of stochastic gene expression.” Physics of life reviews 2.2 (2005): 157-175. online

[5] I. Lestas, J. Paulsson, N. E. Ross and G. Vinnicombe, “Noise in Gene Regulatory Networks,” in IEEE Transactions on Automatic Control, vol. 53, no. Special Issue, pp. 189-200, Jan. 2008, doi: 10.1109/TAC.2007.911347. online

[6] Lyapunov equation. (2023, February 7). In Wikipedia.

[7] Koch, A L, and H R Levy. “Protein turnover in growing cultures of Escherichia coli.” The Journal of biological chemistry vol. 217,2 (1955): 947-57.

[8] Hilfinger, Andreas, and Johan Paulsson. “Separating intrinsic from extrinsic fluctuations in dynamic biological systems.” Proceedings of the National Academy of Sciences 108.29 (2011): 12167-12172. online

[9] Paulsson, J. Summing up the noise in gene networks. Nature 427, 415–418 (2004). online

[10] Fan, Raymond, and Andreas Hilfinger. “The effect of microRNA on protein variability and gene expression fidelity.” Biophysical Journal 122.5 (2023): 905-923. online

[11] Pedraza, J. M., & Paulsson, J. (2008). Effects of molecular memory and bursting on fluctuations in gene expression. Science, 319(5861), 339-343.

[12] Keizer, Joel. Statistical thermodynamics of nonequilibrium processes. Springer Science & Business Media, 2012.

[13] Elf, J., & Ehrenberg, M. (2003). Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome research, 13(11), 2475-2484. online

[14] Huh, D., Paulsson, J. Non-genetic heterogeneity from stochastic partitioning at cell division. Nat Genet 43, 95–100 (2011). online

[15] Hilfinger, A., Norman, T. M., Vinnicombe, G., & Paulsson, J. (2016). Constraints on fluctuations in sparsely characterized biological systems. Physical review letters, 116(5), 058101. online

[16] Risken, H., and T. K. Caugheyz. “The fokker-planck equation: Methods of solution and application.” (1991): 860-860.