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A concentration gradient is the spatial gradient $\nabla c$ of a scalar concentration field. It points toward the greatest local increase of $c$; its components are spatial partial derivatives, so its units are those of concentration divided by length.

For a one-dimensional profile $c(x)=c_0+ax$, the concentration gradient is the constant $a\hat{\mathbf{x}}$. Fick's first law gives a diffusive flux proportional to $-\nabla c$ when that constitutive model applies: the minus sign describes transport toward lower concentration. A fluid velocity can simultaneously advect the constituent in another direction, so a concentration gradient alone does not determine its total flux.

Contact guidance is a directional influence of physical structures, such as aligned fibres or grooves, on cell migration. It can organize trajectories without an imposed soluble concentration gradient.

A diffusion tensor describes direction-dependent diffusive flux. In the local, linear Fickian model,where $C$ is concentration and the entries of $\mathbf{D}$ have the units of a diffusion coefficient. A scalar coefficient is recovered when $\mathbf{D}=D\mathbf{I}$. Material models can instead specify a diagonal matrix or symmetric matrix to represent anisotropic transport.^[ref]

For an original two-dimensional example, chooseA concentration gradient directed along $x$, with components $(g,0)$, produces diffusive flux $(-2g,-g)$ in consistent units. The flux need not be antiparallel to the gradient. Nevertheless,for every nonzero gradient in this positive-definite matrix example.

The eigenvectors $(1,1)/\sqrt{2}$ and $(1,-1)/\sqrt{2}$ have eigenvalues 3 and 1. They identify the principal directions and coefficients. Rotating the coordinate basis changes the matrix entries according to ${\mathbf{D}}^{\prime }=R\mathbf{D}{R}^T$; it does not change these principal coefficients. In an unbounded, homogeneous, drift-free Gaussian model, displacement covariance is $2t\mathbf{D}$, so spreading also differs along those directions.

Symmetry and positive definiteness are assumptions of this ordinary reciprocal diffusion model, not a definition covering every transport law called a tensor. Boundaries also matter: a reflecting boundary condition constrains $\mathbf{n}\cdot \mathbf{D}\nabla C$, which need not be proportional to the normal derivative alone.

GLUT2 is a facilitative glucose transporter with important roles in hepatocytes and in the classical rodent beta-cell model. It supports transport down a concentration gradient and should not be assumed to be the predominant human beta-cell glucose transporter.

Cells store usable free energy in chemical disequilibria: reduced carbon compounds, redox carriers such as NADH, ATP relative to ADP and inorganic phosphate, ion concentration gradients and membrane voltage. These reservoirs are maintained away from equilibrium; the useful quantity is the chemical or electrochemical potential difference, not simply a bond's isolated energy. Some ATP-consuming reactions can be driven because a coupled overall reaction decreases Gibbs free energy, even when the uncoupled synthesis would not.

Oxidation of nutrients transfers reducing equivalents to electron carriers and then to an electron-transport chain. Coupled proton pumping converts part of the redox free-energy drop into a protonmotive force. ATP synthase can convert that force into ATP chemical potential, and ATP hydrolysis drives biosynthesis, transport and mechanical motion. Substrate-level phosphorylation is another ATP source; photosynthetic cells can initially supply the disequilibrium with absorbed light. These are flows with unavoidable dissipation and heat release, not independent energy-creation processes. A reservoir is depleted unless metabolism replenishes it.

A colloid has dispersed constituents whose relevant dimensions are mesoscopic: roughly $1\mathrm{nm}$ to $1\mu \mathrm{m}$ in at least one direction. Interfaces and Brownian motion are consequently important. The size range is a useful convention, not a sharp boundary separating all physical behaviour. A sol of fine silicon dioxide particles in water illustrates solid in liquid; an oil-in-water emulsion illustrates liquid in another immiscible liquid; and a foam illustrates gas dispersed in a liquid, with thin intervening films supplying the small structural dimension. A colloid need not be a thermodynamic equilibrium state.^[ref]

Unprotected particles attract through Van der Waals forces. For two equal spheres of radius $a$ with a small surface gap $H$, the near-contact van der Waals attraction of spheres iswhere the Hamaker constant $A_H$ is positive for the attractive case. The large number of interacting constituent molecules can give a well much deeper than $k_{B}T$, even though individual molecular attractions are weak. Brownian motion brings particles together, and a sufficiently deep attractive well makes separation unlikely. This gives particle aggregation; solid particles can remain distinct within a cluster rather than merging into one continuous solid. Emulsion droplets can additionally coalesce if the separating film ruptures.

Charge stabilization of colloids supplies a repulsive barrier. Like-charged surfaces carry compensating counterions and electric double layers; bringing the layers together costs free energy. DLVO theory combines this repulsion with Van der Waals forces. A barrier large compared with $k_{B}T$ can make aggregation slow, even when sticking would lower the final free energy. Adding salt generally shortens the Debye–Hückel screening length and can lower the barrier; surface charge regulation and specific ion interactions also matter. A negatively charged particle and a positively charged particle need not repel.

Steric stabilization instead coats each particle with adsorbed or grafted polymer chains. In a good solvent, overlap and compression of these layers raise the osmotic pressure and reduce chain entropy, opposing close contact. Electrosteric stabilization combines the two mechanisms. Dense, securely bound layers matter: incomplete adsorption can allow polymer bridging, while nonadsorbing dissolved polymers can create an attractive depletion force. Poor solvent quality can collapse a protective layer. Reducing the density mismatch slows sedimentation but does not remove the interparticle attraction, so uniform suspension and resistance to aggregation are different requirements.

The left panel is an original qualitative potential with arbitrary coefficients: it illustrates a barrier and its reduction by shorter-ranged repulsion, without fitting an experimental DLVO theory curve or extrapolating the continuum attraction to molecular contact. The right panel plots the derived quasi-steady concentration profile in dimensionless variables.

In diffusion-controlled growth, attachment is rapid and delivery through the surrounding medium is the bottleneck. The growing surface depletes the neighboring solute, so a substantial concentration gradient develops. In reaction-controlled growth, arrival is fast compared with incorporation; many arrivals fail to attach, and the surface concentration stays near the reservoir value. The analogous distinction for particle aggregation is between near-unit sticking probability in diffusion-limited cluster aggregation and barrier-suppressed sticking in reaction-limited cluster aggregation. Here the subsequent calculation concerns accretion onto one growing particle, rather than a collision kernel for two mobile clusters.

The printed time-dependent equation is the diffusion equation, not the usual Poisson equation. For constant diffusion coefficient $D$, no bulk sources and negligible advection, Fick's laws give the number flux $\mathbf{j}=-D\nabla c$. Local conservation of mass, expressed here as conservation of the accreting constituent, then givesAn arbitrary time-independent exterior profile does not solve the full moving-boundary problem exactly. The required elementary result uses a quasi-steady approximation: diffusion establishes the exterior profile much faster than the particle radius changes. At each instantaneous radius $a$, set $\partial c/\partial t\simeq 0$. Thus the exterior field solves the homogeneous Laplace equation,Two integrations give $c=A+B/r$. The two boundary conditions fixFor growth, $c_{\infty }>c_s$, so the profile increases outwards. Its radial flux is negative, and the positive inward arrival rate isA material density is needed to turn an arrival rate into a radial velocity. Introduce the constant constituent number density $c_p$ inside the particle, using the same units as $c_s,c_{\infty }$. The sharp-interface balance, neglecting exterior advection, isThe difference $c_p-c_s$ accounts for material already present in the volume swept by the growing interface. In the usual dilute-feed approximation $c_s\ll c_p$, it can be replaced by $c_p$. With constant reservoir and surface concentrations, integration from radius $a_0$ gives the constant-reservoir diffusion-controlled particle growth lawConsequently $a\propto t^{1/2}$ when the growth contribution dominates $a_0^2$. For a finite seed at the literal limit $t\to 0$, the increment $a-a_0$ is instead initially proportional to $t$. Here “early time” means before appreciable reservoir depletion or interactions with other particles, while already beyond the rapid initial establishment of the diffusion field. Indeed, the ratio of diffusion-adjustment time $a^2/D$ to radial-growth time $a/\dot{a}$ is $(c_{\infty }-c_s)/(c_p-c_s)$; small supersaturation relative to the particle density makes the quasi-steady approximation self-consistent. No initial concentration field was specified, so an exact transient solution is not uniquely determined.

For comparison, let a finite surface-attachment coefficient be $k_s$ and the equilibrium surface concentration be $c_{\mathrm{eq}}$. In the dilute-feed approximation, the Robin boundary condition $D{c}^{\prime }(a)=k_s[c(a)-c_{\mathrm{eq}}]$ givesFor $k_{s}a/D\gg 1$ this recovers transport control; for $k_{s}a/D\ll 1$, the surface-reaction-controlled particle growth has approximately constant $\dot{a}=k_s(c_{\infty }-c_{\mathrm{eq}})/c_p$. Thus reaction control gives $a-a_0\propto t$, in contrast to the diffusion-controlled square-radius law. No numerical $c_p$ or $k_s$ is supplied or needed for these scaling results.

The term $\alpha {\nabla }^{2}q$ is cell diffusion arising from random movement. Chemotaxis up the concentration gradient gives the cell flux $\beta q\nabla c$, whose negative divergence is the transport contribution $-\beta \nabla \cdot (q\nabla c)$. The reaction term $q(1-q)$ gives growth governed by the logistic differential equation with dimensionless carrying capacity one.

The chemoattractant itself diffuses, is produced by cells at rate $\kappa \eta q$, and decays at rate $\kappa c$. The homogeneous Neumann boundary conditions suppress the normal diffusive flux of each quantity. They also suppress the normal chemotactic cell flux because the normal derivative of $c$ vanishes. Thus the rectangular boundary is impermeable, and the spatially integrated cell count changes through cell proliferation rather than transport across that boundary.

External chemical signals can guide cell migration, but the statement is too strong if “determined” means that they uniquely specify whether, how fast and in which direction a cell moves. A chemoattractant gradient biases a cell's internal polarity machinery; it does not directly supply the mechanical work of locomotion. Chemotaxis is directional bias, whereas a change in speed or persistence without directional preference is chemokinesis.

A chemical response depends on receptor abundance, signalling adaptation and the cell's existing cell polarity. The same external field can therefore yield different trajectories in different cell states. Uniform stimulation can increase motility without providing a direction, and spontaneous fluctuations of the actin system can initiate movement without a spatial chemical instruction. Persistent migration combines feedback within the cell with a cue that biases that feedback.

Mechanical and geometrical inputs also matter. Durotaxis describes a bias associated with substrate stiffness, while haptotaxis follows spatial variation in adhesive ligands. Contact guidance by fibre alignment can channel motion even when soluble signals are uniform. Narrow spaces constrain the available paths and can force a switch of protrusive mechanism. These effects are not wholly separate from chemistry: cells sense mechanical forces through molecular interactions. Nevertheless, their directional information resides in mechanics or geometry, not necessarily in an imposed soluble concentration gradient.

Contacts between cells supply another source of organization. Experiments on neural crest cells showed that collisions can cause protrusion collapse and repolarization, and that perturbing noncanonical Wnt signalling disrupts this behaviour and group directionality.^[ref] Contact inhibition of locomotion therefore helps generate collective trajectories through local interactions, rather than each cell independently reading a complete external chemical map. Cohesion and other attractive interactions can prevent such repulsion from simply dispersing the group.

A discriminating experiment independently varies soluble gradient, adhesive pattern and mechanical confinement. Measure direction, speed and persistence separately; reverse the gradient, remove it, and then block the relevant receptor while checking viability and the ability to move. If a directional bias follows the reversed chemical field, that supports chemical guidance. If speed rises under uniform stimulation, that supports chemokinesis. If trajectories remain aligned with fibres after chemical reversal, physical guidance is influential. Manipulating intracellular polarity can test whether a cue is instructive or merely permissive.

The useful conclusion is that motion emerges from an internally powered, state-dependent cell interacting with chemical, mechanical and cellular surroundings. External chemical signals are often decisive inputs, but the trajectory is a property of the coupled system.