What Is Mathematical Modeling?

What is a model?¶

Ask how a crawling cell, with no brain, no eyes, and no map, finds its way around a corner, and the answers come quickly: it follows a chemical signal, it senses the wall, it moves randomly and gets lucky, it feels stiffness. Each answer is a proposed mechanism stated informally. Turning such a hypothesis into something that can be tested is what modeling is.

Each part of the definition matters.

• Deliberately simplified. A model is not a copy of reality; a map at 1:1 scale is useless. The simplification is the content of the model, not a regrettable compromise.

• Mathematical language. Ambiguity that survives in English does not survive in an equation. “The cell senses the wall” leaves open how it senses, over what time it integrates, and against what reference it compares.

• A specific question. There is no such thing as the model of a cell, only a model for a question. Change the question and you change the model. Forgetting this is the most common beginner error.

Modeling makes a proposed mechanism precise enough to settle it, true or false, or, when it cannot be settled outright, precise enough to see which mechanisms are even capable of producing the observed behavior. “The cell senses the wall” does neither: it is compatible with every observation, so it can be neither confirmed nor ruled out, and it says nothing about what would suffice. Box’s aphorism, “all models are wrong, but some are useful,” points at the right axis. Wrong is not the interesting question; useful for what is.

Kinds of model¶

Only a mechanistic model can be falsified by an experiment on the mechanism. Fitting a curve well is not evidence for a mechanism.

Successes¶

Despite this, mathematical biology has a long list of successes, built by identifiable people who were often ignored, often working on something else, and occasionally far ahead of their evidence. Each carries its equation, because the equations are the argument.

• Malthus (1798). Population grows geometrically, food arithmetically, with almost no data behind the claim:

  $$\frac{\dd N}{\dd t} = rN \qquad\Longrightarrow\qquad N(t) = N_0\,e^{rt} .$$

  (1)

  $N$ is the number of individuals and $r$ the per-capita growth rate, births minus deaths per individual per unit time. Nothing in the equation can stop the growth. Darwin and Wallace both read Malthus independently and both arrived at natural selection: a crude demographic model triggered the central theory of biology.

• Verhulst (1838). He added the saturation term Malthus lacked and named the result the logistic:

  $$\frac{\dd N}{\dd t} = rN\left(1 - \frac{N}{K}\right) .$$

  (2)

  $K$ is the carrying capacity, the population the environment can support. The new factor is a switch: near $N = 0$ it is one and growth is Malthusian, and it shuts growth off as $N \to K$. The correction was forgotten and rediscovered by Pearl and Reed in 1920.

• Ronald Ross (1911). Having shown that mosquitoes transmit malaria, he did the mathematics and found something no experiment could hand him:

  $$\frac{\dd I}{\dd t} = \beta I(1-I) - \gamma I, \qquad R_0 = \frac{\beta}{\gamma} .$$

  (3)

  $I$ is the infected fraction of the population, $\beta$ the transmission rate, $\gamma$ the recovery rate. The disease grows from rare only if $R_0 > 1$, so control is a threshold and not a total: there is no need to kill every mosquito, only to push the population past the line.

• Lotka (1925) and Volterra (1926). When fishing in the Adriatic largely stopped during the First World War, the proportion of predatory fish went up. Why should less fishing favor sharks?

  $$\frac{\dd V}{\dd t} = aV - bVP - fV, \qquad \frac{\dd P}{\dd t} = c\,bVP - dP - fP .$$

  (4)

  $V$ is prey and $P$ predators; $a$ is prey growth, $b$ the rate at which predators eat prey, $c$ how efficiently eaten prey becomes new predators, $d$ the predator death rate, and $f$ fishing, which kills both. The steady state is the whole answer:

  $$V^\ast = \frac{d+f}{cb}, \qquad P^\ast = \frac{a-f}{b} .$$

  (5)

  Less fishing means smaller $f$, hence fewer prey and more predators. Each species is controlled by the other’s parameters, which intuition does not offer.

• Fisher, Haldane, and Wright (1918 to 1932). They showed that Mendelian genetics and Darwinian selection, widely believed incompatible, were the same theory:

  $$\frac{\dd p}{\dd t} = s\,p(1-p) .$$

  (6)

  $p$ is the frequency of one allele in the population and $s$ its selection coefficient, the fitness advantage it confers. Discrete inheritance and continuous selection fit in one line.

• Hodgkin and Huxley (1952). The squid giant axon is about a millimeter across, wide enough to push a wire down the inside:

  $$C\frac{\dd V}{\dd t} = I - \bar g_{\rm Na}m^3h\,(V - E_{\rm Na}) - \bar g_{\rm K}n^4(V - E_{\rm K}) - \bar g_{\rm L}(V - E_{\rm L}) .$$

  (7)

  $V$ is the voltage across the membrane, $C$ its capacitance, $I$ injected current. Each remaining term is a current through one type of ion channel, with $\bar g$ its maximum conductance and $E$ the voltage at which it reverses. The gating variables $m$, $h$, $n$ are fractions of open channels, each obeying its own ODE, so the model has four equations. Huxley integrated them by hand on a mechanical calculator over roughly three weeks; the work won the 1963 Nobel Prize.

• Turing (1952). “The Chemical Basis of Morphogenesis”:

  $$\partial_t u = D_u\nabla^2 u + f(u,v), \qquad \partial_t v = D_v\nabla^2 v + g(u,v) .$$

  (8)

  $u$ and $v$ are concentrations of two chemicals, $D_u$ and $D_v$ their diffusion coefficients, and $f$, $g$ their reactions. Diffusion, the process that smooths concentrations out, can, when $D_v \gg D_u$, destabilize a uniform state and produce stripes and spots: local activation with long-range inhibition.

• Keller and Segel (1970). Slime mold aggregation, with chemotaxis as an instability:

  $$\partial_t u = \nabla\cdot\bigl(D\nabla u - \chi\,u\nabla c\bigr), \qquad \partial_t c = D_c\nabla^2 c + \alpha u - \beta c .$$

  (9)

  $u$ is the density of cells and $c$ the concentration of the chemical they follow. The first term spreads cells out; the second, with chemotactic sensitivity $\chi$, moves them up the gradient of $c$, which by the second equation they produce themselves at rate $\alpha$. Cells attract cells.

• Purcell (1977). Life at Low Reynolds Number Purcell (1977):

  $$\mathrm{Re} = \frac{\rho u L}{\eta} .$$

  (10)

  $\rho$ is the fluid density, $u$ a speed, $L$ a size, $\eta$ the viscosity. The number compares inertia with viscous drag. For a crawling cell it is about 10^-8 (Example 2), and a bacterium that stops swimming coasts about a tenth of an ångström.

Each of these models has few symbols. A model earns its keep by being small enough to reason about, and every one of them is a deliberate simplification with a domain of validity.

The next page, Doing Some Modeling: Why Inertia Is Irrelevant, sets up the first such simplification, dropping inertia from the motion of a cell, by computing how far a cell coasts; From Newton’s Second Law to a Force Balance makes it and computes exactly what it costs.