Diffusion & Transport in Biology

Fick's First Law from Random Walk

Consider particles on a 1D lattice with spacing $\Delta x$. At each time step$\Delta t$, each particle hops left or right with equal probability $\frac{1}{2}$. The net flux (particles per unit area per unit time) across a boundary at position $x$ is:

Taylor expanding to first order: $c(x \pm \Delta x) = c(x) \pm \Delta x \frac{\partial c}{\partial x} + \cdots$

Identifying the diffusion coefficient $D = \frac{(\Delta x)^2}{2\Delta t}$, we arrive at Fick's first law:

The minus sign ensures that particles flow from high to low concentration. The flux is proportional to the concentration gradient, with $D$ having units of m$^2$/s.

Fick's Second Law from Continuity

Mass conservation requires that the rate of change of concentration equals the negative divergence of the flux. In 1D, the continuity equation is:

Substituting Fick's first law $J = -D \partial c / \partial x$:

This is the diffusion equation (or heat equation). In three dimensions: $\partial c / \partial t = D\nabla^2 c$.

Green's Function: Point Source Solution

For an initial condition of $N$ particles released at the origin at $t=0$, the solution is found by Fourier transform. We seek $c(x,t)$ with$c(x,0) = N\delta(x)$. Taking the Fourier transform in $x$:

The diffusion equation becomes $\partial \hat{c}/\partial t = -Dk^2 \hat{c}$, with solution $\hat{c}(k,t) = N e^{-Dk^2 t}$. Inverting:

This Gaussian spreads with standard deviation $\sigma = \sqrt{2Dt}$. The peak concentration decreases as $1/\sqrt{t}$ in 1D.

Mean Square Displacement

The mean square displacement (MSD) is the key experimental observable. For the Gaussian solution above, we compute:

In $n$ spatial dimensions, each independent coordinate contributes $2Dt$, so:

For 3D diffusion: $\langle r^2 \rangle = 6Dt$. This linear growth with time is the hallmark of normal diffusion. Any deviation from linearity signals anomalous diffusion.

Derivation from Fluctuation-Dissipation

A spherical particle of radius $R$ in a fluid of viscosity $\eta$ experiences Stokes drag $f = 6\pi\eta R v$. The friction coefficient is $\gamma = 6\pi\eta R$. The Langevin equation for such a particle is:

where $\xi(t)$ is the random thermal force with $\langle \xi(t) \rangle = 0$and $\langle \xi(t)\xi(t') \rangle = 2\gamma k_B T \,\delta(t-t')$ (fluctuation-dissipation theorem). Multiplying by $x$ and taking the ensemble average, we compute$\frac{d}{dt}\langle xv \rangle$:

Using the equipartition theorem $m\langle v^2 \rangle = k_BT$ and noting that in steady state $\langle xv \rangle = \frac{1}{2}\frac{d}{dt}\langle x^2 \rangle$, for times $t \gg m/\gamma$ (overdamped limit):

Therefore the diffusion coefficient is $D = k_BT/\gamma$, and substituting the Stokes friction coefficient:

This is the Stokes-Einstein relation. It connects microscopic diffusion to macroscopic viscosity. For a typical protein ($R \sim 3$ nm) in water at 25$^\circ$C: $D \approx 70$ $\mu$m$^2$/s.

Non-Spherical Particles: Ellipsoids

For non-spherical particles, the friction coefficient depends on the shape and orientation. For a prolate ellipsoid (cigar-shaped) with semi-axes$a > b = c$ and aspect ratio $p = a/b$, the translational friction coefficient along the long axis is:

And perpendicular to the long axis:

For an oblate ellipsoid (disk-shaped) with $a = b > c$and aspect ratio $p = a/c > 1$, the orientation-averaged friction coefficient is:

In biology, DNA behaves as an elongated polymer and proteins are often non-spherical. The Perrin factors $f/f_0$ (ratio to an equivalent sphere) typically range from 1.0 to 1.4 for globular proteins and can exceed 2.0 for fibrous proteins.

General Anomalous Diffusion

The generalized MSD for anomalous diffusion is:

where $D_\alpha$ is the generalized diffusion coefficient (with units m$^2$/s$^\alpha$) and $\alpha$ is the anomalous exponent:

• • $\alpha = 1$: Normal (Fickian) diffusion
• • $\alpha < 1$: Subdiffusion — particles are transiently trapped
• • $\alpha > 1$: Superdiffusion — particles have persistent directed motion

Subdiffusion from Crowding and Caging

Subdiffusion ($\alpha < 1$) arises from the continuous-time random walk (CTRW) model, where waiting times between jumps follow a heavy-tailed distribution $\psi(\tau) \sim \tau^{-(1+\alpha)}$ for large $\tau$. The mean waiting time diverges: $\langle \tau \rangle = \infty$.

In the CTRW framework, the probability of not having jumped by time $t$ is the survival function $\Psi(t) = 1 - \int_0^t \psi(\tau)\,d\tau \sim t^{-\alpha}$. The MSD becomes:

In cells, subdiffusion is commonly observed for proteins ($\alpha \approx 0.7$) and mRNA ($\alpha \approx 0.4{-}0.7$) due to molecular crowding by macromolecules that occupy 20-40% of the cell volume, creating a cage-like environment.

Superdiffusion from Active Transport

Superdiffusion ($\alpha > 1$) in cells arises from motor-driven transport along cytoskeletal filaments. A particle alternating between diffusion (rate $D$) and directed motion at velocity $v$ with switching rates $k_{\text{on}}$ and$k_{\text{off}}$ has MSD:

At short times ($t \ll 1/k_{\text{off}}$), the ballistic term dominates:$\langle x^2 \rangle \approx v^2 t^2$ giving $\alpha = 2$. At long times, it crosses over to enhanced diffusion: $\langle x^2 \rangle \sim (2D + v^2/k_{\text{off}})t$, returning to $\alpha = 1$ but with an augmented diffusion coefficient.

The Fractional Diffusion Equation

The CTRW with heavy-tailed waiting times leads to a fractional diffusion equation. Starting from the CTRW master equation in Fourier-Laplace space and taking the continuum limit, the probability density $p(x,t)$ satisfies:

where $ {}_0D_t^{1-\alpha}$ is the Riemann-Liouville fractional derivative of order$1 - \alpha$:

The fractional derivative introduces memory into the diffusion process: the current flux depends on the entire history of the concentration field. For$\alpha = 1$, the ordinary diffusion equation is recovered. The Green's function of this equation is a Fox H-function, which decays as a stretched Gaussian$\sim \exp(-c \cdot x^{2/(2-\alpha)})$ for large $x$.

Smoluchowski Rate for Diffusion-Limited Binding

Consider a perfectly absorbing sphere of radius $R$ immersed in a solution of particles with concentration $c_0$ far from the sphere and diffusion coefficient $D$. In steady state, the concentration satisfies $\nabla^2 c = 0$ with boundary conditions$c(R) = 0$ and $c(\infty) = c_0$. In spherical coordinates:

The general solution is $c(r) = A + B/r$. Applying boundary conditions:$c(r) = c_0\left(1 - R/r\right)$. The flux at the surface is:

The total rate (particles per unit time) is $J(R) \times 4\pi R^2$:

This is the Smoluchowski rate. For two diffusing species with mutual diffusion coefficient $D = D_1 + D_2$ and encounter distance $R = R_1 + R_2$, the bimolecular rate constant is $k = 4\pi(D_1 + D_2)(R_1 + R_2)$. In molar units, this gives $k \sim 10^9{-}10^{10}$ M$^{-1}$s$^{-1}$.

Berg-Purcell Limit: Receptors on a Sphere

Real cells do not absorb ligands uniformly. Instead, $N$ small receptors of radius$a$ are distributed on a spherical cell of radius $R$. Berg and Purcell (1977) showed that the effective capture rate is:

The reduction factor compared to a fully absorbing sphere is:

Derivation of the reduction factor: Each receptor on the sphere acts as a small absorbing disk. Far from the sphere, the concentration field is approximately spherically symmetric. Near each receptor, the local problem is diffusion to a disk of radius $a$ on an otherwise reflecting surface, giving a local capture rate of $4Da$ per receptor. The total rate for $N$ independent receptors would be $4NDa$, but this overcounts because each receptor depletes the local concentration. The effective concentration seen by each receptor is reduced to $c_{\text{eff}} = c_0 \cdot \pi R/(Na + \pi R)$, giving the combined rate above.

The remarkable result is that even 1% surface coverage ($Na/\pi R \sim 0.01$) gives ~1% of the maximum rate, but due to the sublinear scaling, 10,000 receptors with 0.01% coverage can capture at over 50% of the Smoluchowski limit. Cells exploit this to sense their chemical environment with extraordinary sensitivity.

The Target Search Problem

A transcription factor (TF) must find its specific binding site (typically ~20 bp) on a genome of millions of base pairs. The lac repressor in E. coli finds its operator in about 6 minutes, much faster than pure 3D diffusion would predict (~60 minutes). The resolution comes from facilitated diffusion: alternating between 1D sliding along DNA and 3D hopping through solution.

Derivation of the Optimal Search Time

Let $D_1$ be the 1D sliding diffusion coefficient along DNA and $D_3$ be the 3D diffusion coefficient. The protein slides for a length $\ell$ during each encounter with DNA, taking time $\tau_{\text{slide}} = \ell^2/(2D_1)$. Between encounters, the protein diffuses in 3D for time $\tau_{\text{hop}}$.

The total genome of length $L$ must be covered. The number of sliding events needed to scan the entire genome is $n = L/\ell$ (assuming non-redundant scanning). The total search time is:

The hopping time depends on the DNA concentration in the cell:$\tau_{\text{hop}} \approx 1/(4D_3 c_{\text{DNA}})$ where $c_{\text{DNA}} = L/V_{\text{cell}}$is the DNA line density per unit volume. Minimizing $t_{\text{search}}$ with respect to $\ell$:

Solving for the optimal sliding length:

Substituting back, the minimum search time is:

Speed-Up Factor vs Pure 3D Diffusion

For pure 3D diffusion to a target of size $a$ in volume $V$, the mean search time is $t_{3D} = V/(4\pi D_3 a)$. The speed-up factor from facilitated diffusion is:

For E. coli parameters ($L \approx 1.5$ mm, $V \approx 1$ $\mu$m$^3$,$D_1 \approx 0.05$ $\mu$m$^2$/s, $D_3 \approx 3$ $\mu$m$^2$/s), the speed-up is approximately 100-fold. This explains how proteins can locate their target sequences within minutes despite the vast genome size. The optimal sliding length is typically $\ell_{\text{opt}} \sim 50{-}200$ bp, consistent with single-molecule experiments.