Systems Biophysics

Derivation: The Hill Function for Gene Regulation

Consider a gene whose transcription rate depends on the concentration of an activating transcription factor [TF]. If $n$ TF molecules must bind cooperatively to the promoter to activate transcription, the binding equilibrium is:

$$n \cdot \text{TF} + \text{DNA} \rightleftharpoons \text{TF}_n\text{-DNA}$$

The equilibrium dissociation constant is $K^n = [\text{TF}]^n[\text{DNA}] / [\text{TF}_n\text{-DNA}]$. The fraction of promoters occupied is:

$$f = \frac{[\text{TF}]^n}{K^n + [\text{TF}]^n}$$

The protein production rate is the sum of basal and activated transcription:

$$\boxed{[\text{protein}] = \alpha_0 + \alpha \frac{[\text{TF}]^n}{K^n + [\text{TF}]^n}}$$

where $\alpha_0$ is the basal rate and $\alpha$ is the maximum activated rate. This is the Hill function. The Hill coefficient $n$ controls the steepness of the response:

• $n = 1$: Michaelis-Menten (hyperbolic) response — no cooperativity
• $n = 2$: Moderate cooperativity, as in hemoglobin O$_2$ binding
• $n \to \infty$: Step function (digital switch)

Ultrasensitivity ($n > 1$): The input concentration range required to go from 10% to 90% activation is:

$$\frac{[\text{TF}]_{90}}{[\text{TF}]_{10}} = 81^{1/n}$$

For $n = 1$, this ratio is 81 (a gradual response); for $n = 4$, it is $81^{1/4} = 3$ (a sharp, switch-like response).

Signal-to-noise ratio: For a Hill function response, the sensitivity (gain) is $g = d[\text{protein}]/d[\text{TF}]$. At the midpoint ($[\text{TF}] = K$), the maximum gain is:

$$g_{\max} = \frac{\alpha n}{4K}$$

Higher cooperativity amplifies the signal relative to input noise, improving the signal-to-noise ratio by a factor of $n$.

Bistability from Positive Feedback

A gene that activates its own expression (autoactivation) with cooperativity can exhibit bistability — two stable steady states (ON and OFF) for the same parameter values. The steady-state equation for a self-activating gene with degradation rate $\gamma$ is:

$$\frac{d[P]}{dt} = \alpha \frac{[P]^n}{K^n + [P]^n} - \gamma [P] = 0$$

Graphically, the production rate (sigmoidal) and degradation rate (linear) can intersect at one or three points. Three intersections give two stable states (low and high expression) separated by an unstable intermediate. The condition for bistability requires $n \geq 2$and a sufficiently large $\alpha/(\gamma K)$ ratio.

Derivation: Keller-Segel Equations for Chemotaxis

At the population level, the density of cells $\rho(\mathbf{x}, t)$ in a chemoattractant field $c(\mathbf{x}, t)$ evolves according to the Keller-Segel equations:

$$\boxed{\frac{\partial\rho}{\partial t} = D\nabla^2\rho - \chi\nabla\cdot(\rho\nabla c)}$$

The first term is diffusion (random motility with coefficient $D$). The second term is chemotactic drift: cells move up the gradient of $c$ with chemotactic sensitivity $\chi$.

Deriving $\chi$ from receptor-ligand binding:The chemotactic response depends on the fraction of receptors bound to ligand. For a receptor with dissociation constant $K_d$:

$$f(c) = \frac{c}{K_d + c}$$

The chemotactic sensitivity is proportional to the derivative of receptor occupancy:

$$\chi = \chi_0 \frac{df}{dc} = \chi_0 \frac{K_d}{(K_d + c)^2}$$

This predicts maximum sensitivity at low ligand concentrations ($c \ll K_d$) and decreased sensitivity at saturation ($c \gg K_d$).

The chemoattractant may itself be produced, consumed, or diffuse:

$$\frac{\partial c}{\partial t} = D_c\nabla^2 c + f(c, \rho)$$

where $f(c, \rho)$ represents production/consumption. A famous result is that the Keller-Segel system can exhibit chemotactic collapse: if $\chi/D$ exceeds a critical threshold, cells aggregate into singular concentrations (blow-up in finite time in 2D), modeling the formation of bacterial colonies.

Derivation: E. coli Run-and-Tumble Drift Velocity

An E. coli cell swims at speed $v \approx 20$ $\mu$m/s during runs (average duration $\tau_{\text{run}} \approx 1$ s) and randomly reorients during tumbles (duration $\tau_{\text{tumble}} \approx 0.1$ s). The tumble bias (fraction of time tumbling) is modulated by the chemotaxis signaling pathway.

In a gradient, runs up the gradient are extended (lower tumble rate) while runs down the gradient are shortened. If the tumble rate is $\lambda(\theta)$ where $\theta$is the angle between the swimming direction and the gradient, and the run length increases by a factor $(1 + \beta\cos\theta)$ for small biases:

$$\tau_{\text{run}}(\theta) = \tau_0(1 + \beta\cos\theta)$$

The drift velocity along the gradient direction, averaged over all orientations (in 3D):

$$v_{\text{drift}} = \frac{v\beta}{3}$$

where the factor 1/3 comes from averaging $\cos^2\theta$ over a sphere. Typically $\beta \sim 0.1\text{--}0.3$, giving drift velocities of$\sim 1\text{--}5$ $\mu$m/s, much less than the swimming speed but sufficient for efficient navigation over cell-length scales.

Derivation: Turing Instability Conditions

Consider a two-component reaction-diffusion system with concentrations $u(\mathbf{x},t)$(activator) and $v(\mathbf{x},t)$ (inhibitor):

$$\frac{\partial u}{\partial t} = D_u\nabla^2 u + f(u,v)$$

$$\frac{\partial v}{\partial t} = D_v\nabla^2 v + g(u,v)$$

Let $(u_0, v_0)$ be a spatially uniform steady state: $f(u_0, v_0) = 0$,$g(u_0, v_0) = 0$. Linearize about this state with perturbations$\tilde{u}, \tilde{v} \propto e^{\sigma t + i\mathbf{k}\cdot\mathbf{x}}$:

$$\begin{pmatrix} \sigma\tilde{u} \\ \sigma\tilde{v} \end{pmatrix} = \begin{pmatrix} f_u - D_u k^2 & f_v \\ g_u & g_v - D_v k^2 \end{pmatrix} \begin{pmatrix} \tilde{u} \\ \tilde{v} \end{pmatrix}$$

where $f_u = \partial f/\partial u|_0$, etc., are elements of the Jacobian, and$k = |\mathbf{k}|$ is the wavenumber. The growth rate $\sigma(k)$satisfies the characteristic equation:

$$\sigma^2 - \sigma[\text{tr}(J_k)] + \det(J_k) = 0$$

where $J_k$ is the matrix above. For Turing instability, we need the uniform state to be stable without diffusion but unstable with diffusion:

Condition 1 (stable without diffusion, $k=0$):

$$f_u + g_v < 0 \quad \text{and} \quad f_u g_v - f_v g_u > 0$$

Condition 2 (unstable with diffusion for some $k > 0$):We need $\det(J_k) < 0$ for some $k$:

$$D_v f_u + D_u g_v > 2\sqrt{D_u D_v(f_u g_v - f_v g_u)}$$

This requires $D_v f_u + D_u g_v > 0$. Since the trace condition gives$f_u + g_v < 0$, we need $f_u > 0$ (activator self-activates) and$g_v < 0$ (inhibitor self-inhibits) with $D_v \gg D_u$ (the inhibitor must diffuse much faster than the activator). The canonical requirement is:

$$\boxed{d = \frac{D_v}{D_u} \gg 1 \quad \text{(long-range inhibition, short-range activation)}}$$

Critical wavenumber: The most unstable wavenumber$k_c$ (fastest growing mode) is found by minimizing $\det(J_k)$ with respect to $k^2$:

$$k_c^2 = \sqrt{\frac{f_u g_v - f_v g_u}{D_u D_v}}$$

The resulting pattern wavelength is:

$$\boxed{\lambda_c = \frac{2\pi}{k_c} = 2\pi\left(\frac{D_u D_v}{f_u g_v - f_v g_u}\right)^{1/4}}$$

Derivation: Lotka-Volterra Predator-Prey System

Let $x(t)$ be the prey population and $y(t)$ the predator population. The Lotka-Volterra equations are:

$$\frac{dx}{dt} = \alpha x - \beta xy \quad \text{(prey: growth minus predation)}$$

$$\frac{dy}{dt} = \delta xy - \gamma y \quad \text{(predator: conversion minus death)}$$

Fixed points: (1) Trivial: $(x^*, y^*) = (0, 0)$ (unstable saddle); (2) Coexistence: $(x^*, y^*) = (\gamma/\delta, \alpha/\beta)$ (center — neutrally stable).

Conserved quantity: The Lotka-Volterra system has a conserved integral of motion. Divide the two equations:

$$\frac{dy}{dx} = \frac{\delta xy - \gamma y}{\alpha x - \beta xy} = \frac{y(\delta x - \gamma)}{x(\alpha - \beta y)}$$

Separating variables: $\frac{\alpha - \beta y}{y} dy = \frac{\delta x - \gamma}{x} dx$. Integrating both sides:

$$\alpha\ln y - \beta y = \delta x - \gamma\ln x + C$$

Rearranging:

$$\boxed{H(x, y) = \delta x - \gamma\ln x + \beta y - \alpha\ln y = \text{const}}$$

This conserved quantity means the orbits in $(x, y)$ phase space are closed curves around the coexistence fixed point. The populations oscillate periodically with the period depending on initial conditions. The system has no limit cycle; all solutions are periodic orbits.

Logistic growth and competitive exclusion: The logistic equation$dN/dt = rN(1 - N/K)$ incorporates a carrying capacity $K$. For two competing species:

$$\frac{dN_1}{dt} = r_1 N_1 \left(1 - \frac{N_1 + \alpha_{12}N_2}{K_1}\right)$$

$$\frac{dN_2}{dt} = r_2 N_2 \left(1 - \frac{N_2 + \alpha_{21}N_1}{K_2}\right)$$

The competitive exclusion principle (Gause): stable coexistence requires $\alpha_{12} < K_1/K_2$ AND $\alpha_{21} < K_2/K_1$ — each species must inhibit itself more than it inhibits the other. Otherwise, one species drives the other to extinction.

Derivation: Intrinsic vs Extrinsic Noise

The total variance in protein expression across a population has two contributions:

$$\sigma_{\text{total}}^2 = \sigma_{\text{int}}^2 + \sigma_{\text{ext}}^2$$

Intrinsic noise ($\sigma_{\text{int}}^2$): arises from the stochastic nature of individual biochemical reactions (random binding of polymerase, stochastic mRNA production and degradation). It is specific to each gene copy.

Extrinsic noise ($\sigma_{\text{ext}}^2$): arises from fluctuations in shared cellular components (ribosomes, RNA polymerase, cell volume, growth rate). It affects all genes in a cell equally.

Two-reporter decomposition (Elowitz et al., 2002): Express two identical but distinguishable reporters (e.g., CFP and YFP) from identical promoters in the same cell. Let $x_1$ and $x_2$ be their expression levels:

$$\sigma_{\text{int}}^2 = \frac{1}{2}\langle(x_1 - x_2)^2\rangle$$

$$\sigma_{\text{ext}}^2 = \langle x_1 x_2\rangle - \langle x_1\rangle\langle x_2\rangle$$

The logic: when both reporters are high or both are low (correlated), the fluctuation is extrinsic (shared cause). When one is high and the other low (anticorrelated), the fluctuation is intrinsic (independent stochastic events).

The noise strength is conventionally measured by the squared coefficient of variation:

$$\eta^2 = \frac{\sigma^2}{\langle x\rangle^2} = \eta_{\text{int}}^2 + \eta_{\text{ext}}^2$$

Derivation: Fano Factor and Transcriptional Bursting

For a simple birth-death process (constitutive production at rate $k$, degradation at rate $\gamma$), the steady-state distribution is Poisson with mean$\langle n\rangle = k/\gamma$ and:

$$\text{Fano factor} = F = \frac{\sigma^2}{\langle n\rangle} = 1 \quad \text{(Poisson)}$$

In reality, gene expression occurs in bursts: the promoter switches stochastically between ON and OFF states, and during each ON period, multiple mRNAs are produced. The three-stage model:

$$\text{Gene}_{\text{OFF}} \underset{k_{\text{off}}}{\overset{k_{\text{on}}}{\rightleftharpoons}} \text{Gene}_{\text{ON}} \xrightarrow{k_m} \text{mRNA} \xrightarrow{k_p} \text{Protein}$$

In the limit of fast promoter switching relative to mRNA lifetime ($k_{\text{on}}, k_{\text{off}} \gg \gamma_m$), the mRNA is produced in geometrically distributed bursts of mean size:

$$b = \frac{k_m}{k_{\text{off}}}$$

Each mRNA burst is translated into a burst of proteins. The protein burst sizeis $b_p = k_p / \gamma_m$ (mean number of proteins per mRNA). The resulting Fano factor for protein number is:

$$\boxed{F = 1 + b_p = 1 + \frac{k_p}{\gamma_m}}$$

More generally, when both transcriptional and translational bursting contribute:

$$F = 1 + b \quad \text{where } b = b_{\text{transcription}} \times b_{\text{translation}}$$

Experimental measurements in E. coli and yeast show $F \approx 2\text{--}10$(super-Poissonian), consistent with burst sizes of 1–9 proteins. In mammalian cells,$F$ can be much larger due to infrequent, large transcriptional bursts.