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# Are theoretical results ’Results’?

Raymond E. Goldstein

###### Abstract

Yes.

##  1 Introduction

The decision letter from the journal was very supportive - it was clear our
paper (Kirkegaard et al., 2016) would be published - but one of the referees
definitely did not like the way we had combined experimental biology and
physical calculations in our paper: "The data should be described and the
inferences drawn, and the modelling relegated to its proper place as
quantitative verification of the inferences that can be made directly from the
data."

And this was not an isolated case; a referee of another paper had said:
"Instead, the authors should let the data speak for itself, and postpone
heavier theoretical analysis for later, perhaps in the Discussion." Many of my
colleagues have experienced the same reaction to papers mixing theory and
experiment. What were we doing wrong? Why was it not OK, according to these
referees, to present the observations and the theory in a back-and-forth
dialogue within the ’Results’ section?

While I was bemused by these statements (relegated!), they resonated with my
long experience with some in the biology community, namely that they see the
significance of theory very differently from the way physicists understand it.
For many biologists, theoretical results are simply not ’Results’. Indeed, I
suspect to many they are seen as a matter of opinion, without any intrinsic
significance. In essence, they don’t add anything new. Hence the belief in the
canonical Results/Discussion dichotomy in which theory (or ’modelling’, as it
is often called) plays second fiddle, or third.

In contrast, physicists are brought up to think by means of mathematical
models: harmonic oscillators, random walks, idealized electrical circuits and
so on are among the tools in our toolbox, whether we do experiment or theory.
We use them as solvable examples in which a well-defined set of assumptions
leads to precise outcomes, and where the dependence of the outcomes on all of
the parameters can be interpreted. This approach allows us to estimate what is
important and what is not in any setting. Models also help us to think about
problems: "If this is the underlying physics, then A should vary with B
quadratically…", or "under these assumptions, the data should collapse like
this…" or, when we spot something is not quite right, "here I argue that these
claims are in conflict with basic laws of physics" (Meister, 2016).

The role of theory is also intimately connected with predictions. While I know
biologists who would say “who cares about a prediction in the absence of
experiment?”, physicists are brought up to celebrate them - they are the stuff
of legend, from Dirac’s prediction of antiparticles and Einstein’s prediction
of the bending of starlight, to the work by many that predicted the Higgs
particle. We view predictions as motivations for experiment and as a means to
move the discipline forward. Of course, sometimes they turn out to be wrong,
but that is often how science works. Even if theoretical work does not take
the form of a prediction, per se, it may still be very useful to design
experiments with theory in mind, as emphasized by Bialek (2018), who has
described many historical examples of the role theory has played in biology,
from Rayleigh’s work on hearing to Watson and Crick.

My purpose here is to push back against the view that theory is not a
’Result’. I argue for the unabashed inclusion of mathematical formulations and
pedagogy within the body of papers published in eLife and other primarily
biological journals. By interleaving the experimental and theoretical results
it is possible to tell a story, and I firmly believe this makes for much more
interesting and readable papers. It is also faithful to the scientific method,
in which one goes back and forth with experiment and hypothesis.

Readers may be interested to learn that biological information, background and
results are now routinely included in papers published in physics journals,
although this has not always been the case: I vividly recall a situation
several decades ago when a colleague, a high-energy physicist, saw a preprint
about pattern formation in the slime mold Dictyostelium discoideum on my desk
and asked: "Why would any physicist study something as ridiculous as that?"
But by now many physicists do exactly that, and many physics journals are full
of discussions of cAMP signaling, spiral waves, and chemotaxis (Goldstein,
1996; Rappel et al., 1999; Gholami et al., 2015). If we really take
interdisciplinary research seriously then I assert there has to be a prominent
place for theory within biology papers, both as Results in papers that combine
experiment and theory, and as Results in theory papers.

This is nothing new. If you have not already done so, I highly recommend
reading the celebrated paper by Hodgkin and Huxley (1952) to see experiments
and theory interleaved. Theory is not relegated to the discussion, or worse,
to supplementary material, but instead is incorporated into the body of the
paper as if it is the most natural thing to do. And this was in the Journal of
Physiology. The same structure is found in the Michaelis-Menten paper, which
was published (in German) in a biochemistry journal (Michaelis and Menten,
1913); (Johnson and Goody, 2011). If this was appropriate a century ago, why
must details of mathematical models now be relegated to the back of papers
(see, for example, Paulick et al. (2017), Ferreira et al. (2017), and
Streichan et al. (2018))?

Many readers will appreciate that the issue I am raising about quantitative
descriptions of living systems is closely associated with the tension that
exists between the stereotypes of the biologist, who wants to incorporate all
the complexity of a particular system, and the physicist who seeks generality
and minimalism. As has been emphasized in other recent opinion pieces (Shou et
al., 2015; Riveline and Kruse, 2017), the role of theory in biology has been
growing and this development requires new ways of training scientists on both
sides of the physics/biology divide. Less attention has been paid to providing
concrete examples for the biology community of how physicists think about
understanding data, and this essay’s goal, in part, is to address this lacuna.

Well aware of the risks of trying to speak for an entire community, below I
take the reader through an example of how (at least some) physicists might go
about describing a well-known phenomenon that shows up everywhere in biology -
from the functioning of cellular receptors to bacterial chemotaxis, the
propagation of action potentials, and fluorescence recovery after
photobleaching (FRAP) - namely, diffusion. Employing poetic license, I imagine
that we are at a point in time when the diffusion equation itself was not
known, nor was Fick’s Law, so both the experimental observations and
theoretical analysis presented below are new and worthy of being described as
Results.

I compose two versions of a Results section to indicate various ways of
presenting the data and theory interleaved in a compact presentation that (I
hope) is widely understandable by the community. The first version involves a
’microscopic’ model that is a caricature of the biological system, but
contains the essential ingredients to display the behavior observed on the
large scale. The way in which microscopic parameters enter into the
macroscopic answer turns out to be general (or, as physicists say,
’universal’), a key take-home lesson. The second version - which is probably
more challenging - involves the use of ’dimensional analysis’, one of the most
powerful methods of analyzing natural phenomena. Here, relationships between
various quantities are deduced by examining the units in which they are
measured (mass, length, time, charge, etc.). Introduced long ago, particularly
in the work of Maxwell (1869), this technique can often lead to exact answers
to problems, up to the proverbial ’factors of two’.

##  2 A discovery

Figure 1: Experimental setup to study diffusion in the green alga
Chlamydomonas. (a) A light sheet is used to gather the algae, which are
swimming in a petri dish, into a narrow strip of cells along the $y$-axis. (b)
After the light is turned off, the cells swim randomly and spread out. The
concentration profile, $C(x,t)$, is then measured along a thin strip parallel
to the $x$-axis; $t$ is time.

Allow me to introduce our fictitious Professor Lamarr, who has been
investigating how the single-cell green alga Chlamydomonas move in response to
light. She has discovered that if a narrow sheet of light is directed into an
algal suspension in a petri dish (fig1a), the algae swim into the beam and
form a concentrated line of cells. When the light is turned off and there is
no more phototactic cue, the cells resume a random swimming motion described
previously (Polin et al., 2009), in which every $10$ s or so their roughly
linear motion is interrupted by a turn: the angle of this turn falls within a
distribution that has a mean of $\sim\\!90$ degrees. These random turns lead
the population to spread out over time (fig1b). See Methods for experimental
details.

Lamarr measures the normalized concentration profiles, $C(x,t)$, in a thin
strip that is perpendicular to the initial line of cells, obtaining the data
shown in fig2a. The sharply-peaked profile at early times gradually spreads
out until the Petri dish is uniformly filled with cells. She measured the
variance $\langle x^{2}\rangle$ of the concentration profile, and found the
linear relation $\langle x^{2}\rangle={\mathscr{D}}t$, with
${\mathscr{D}}=0.2$ mm2/s (fig2b). Finally, the peak height $C(0,t)$ decays
smoothly with time (fig2c). By systematic experimentation, she found that the
basic results were insensitive to the precise size of the initial gathering,
and that various swimming mutants of Chlamydomonas displayed the same
behavior, albeit with different values of ${\mathscr{D}}$.

Figure 2: Experimental results on diffusion in a population of the green alga
Chlamydomonas. (a) Concentration profiles, $C(x,t)$, normalized to unity, at
the following times: 1 second (red), 3 seconds (green), 7 seconds (blue) and
30 seconds (black). (b) The variance, $\langle x^{2}\rangle$, of the data
shown in (a) as a function of time; the dashed magenta line is a linear fit to
the data. (c) The peak height, $C(0,t)$, of the data shown in (a) as a
function of time.

###  2.1 Results v1: Experimental observations explained by a microscopic
model

In this version of Results, we begin with a theoretical model of the random
motions of individual cells and deduce from it a population-level description
with which to analyze the data. In the simplest picture, we assume that cells
move only to the left and right along the $x$-axis, and the cells are
constrained to sit on a discrete set of points, at positions $x_{m}=m\Delta$,
where $m=1,2,3,\ldots$ (fig3a). Likewise, we assume time is discrete, so at
each time $t_{n}=n\tau$, $n=1,2,3,\ldots$, a cell moves with probability $1/2$
to the left or right, as indicated by the arrows in fig3a.

Figure 3: A random walk in one dimension. (a) A cell at site $m$ moves with
probability $1/2$ to the left or right. (b) Diagram illustrating the counting
that underlies the evolution equation (1).

In order to find an evolution equation for the probability $C_{n}(m)$ of
finding a cell at position $m\Delta x$ at time $n\Delta t$ we observe (fig3b)
that cells that appear at point $m$ at time $n+1$ arrived there by moving to
the right from point $m-1$ or by moving to the left from point $m+1$ at the
previous time step (each with probability $1/2$). Thus we can deduce that

$C_{n+1}(m)=\frac{1}{2}C_{n}(m+1)+\frac{1}{2}C_{n}(m-1).$ (1)

We now imagine that the probabilities are varying sufficiently slowly in space
and time that we can use the following Taylor expansions: $C_{n+1}(m)\simeq
C_{n}(m)+\tau(\partial C_{n}(m)/\partial t)+\cdots$; and $C_{n}(m\pm 1)\simeq
C_{n}(m)\pm\Delta(\partial C_{n}(m)/\partial
x)+(\Delta^{2}/2)(\partial^{2}C_{n}(m)/\partial x^{2})+\cdots$. Collecting
terms, we deduce that the ’continuum limit’ for this one-dimensional random
walk is

$\frac{\partial C}{\partial t}=D\frac{\partial^{2}C}{\partial
x^{2}}\leavevmode\nobreak\ ,\ \ \ \ \ {\rm with}\ \ \ \ \
D=\frac{\Delta^{2}}{2\tau}.$ (2)

We term this the ‘diffusion equation’, where the diffusion constant $D$ has
units of length2/time. Although the above was derived in the context of a
model with discrete space and time coordinates, the crucial point is that we
can more generally interpret $\Delta$ as the typical distance a cell travels
between sharp turns, and $\tau$ as the time between such turns. If $U$ is the
swimming speed between turns, then $\Delta\sim U\tau$, so we can write
$D=U^{2}\tau/2$. From tracking studies of Chlamydomonas, we know that $U\sim
0.1$ mm/s, and $\tau\sim 10$ s, and therefore $\Delta\sim 1$ mm and $D\sim
0.1$ mm2/s.

If we rewrite the diffusion equation (2) as $\partial C/\partial
t=-(\partial/\partial x)(-D\partial C/\partial x)$ then it can be written as

$\frac{\partial C}{\partial t}=-\frac{\partial J}{\partial x},\ \ \ \ {\rm
where}\ \ \ \ J=-D\frac{\partial C}{\partial x},$ (3)

where we identify the flux $J$ as the number of cells passing through a given
point $x$ per unit time. This relationship implies that cells pass from
regions of high concentration to low at a rate proportional the gradient of
concentration. This ’flux form’ of the diffusion equation guarantees that the
total number of cells, $N=\int_{-\infty}^{\infty}dxC(x,t)$, remains constant
over time, since

$\frac{dN}{dt}=\int_{-\infty}^{\infty}dx\frac{\partial C(x,t)}{\partial
t}=-\int_{-\infty}^{\infty}dx\frac{\partial J}{\partial
x}=J(-\infty)-J(+\infty).$ (4)

Thus, provided the flux $J$ goes to zero far away from our point of
observation, $N$ is constant.

The relationship (Fick’s Law) $J=-D\partial C/\partial x$ can be tested
experimentally. Lamarr recorded the distributions of cells at the times
indicated in fig2 and then again $0.2$ s later. As shown in fig4a for one
pair, such measurements yield the flux, $J$, and concentration gradient,
$\partial C/\partial x$ each as functions of $x$ (fig4b), and we see that,
apart from the overall scale, they are oppositely signed, as predicted by (3).
But we can now go one step further and plot $J$ at each point $x$ and time $t$
versus $\partial C/\partial x$ at those same $x$ and $t$ values. If the theory
is correct, then every data set should collapse on to a single straight line,
and indeed this is the case (fig4c). According to the theory above, the slope
of the line in fig4c is the diffusion constant $D$; we obtain $D=0.1$ mm2/s,
which is consistent with the microscopic interpretation in terms of motility.

Figure 4: Flux and the diffusion equation. (a) Concentration profiles,
$C(x,t)$, at times $t=3$ and $t=3.2$ s . (b) The flux of cells past a given
point, $J$ (black; left axis), and the concentration gradient, $\partial
C/\partial x$ (yellow; right axis), versus position, $x$. (c) Flux, $J$,
versus concentration gradient, $\partial C/\partial x$, for all the values of
$x$ and $t$ shown in fig2a. The dashed magenta line has a slope $D=0.1$ mm2/s.

###  2.2 Results v2: Dimensional analysis leads to the diffusion equation

In this version of the Results section our goal is to infer directly from the
data a differential equation for the time evolution of the algal concentration
$C(x,t)$, which is measured in organisms per mm, hence units of 1/length. The
variance $\langle x^{2}\rangle$ has, of course, units of length squared, so we
can define a characteristic, time-dependent length $\ell(t)=\sqrt{\langle
x^{2}\rangle}$. From the fit to the data in fig2b we infer that the width of
$C(x,t)$ grows as

$\ell(t)\sim\sqrt{{\mathscr{D}}t}.$ (5)

A very natural question is whether $\ell(t)$ is the only intrinsic length
scale that can be extracted from the data. As $C(x,t)$ has units of
number/length we can, without loss of generality, write
$C(x,t)=\ell(t)^{-1}F(x,t)$ for some unknown function $F$ that is itself
dimensionless. And since $F$ is dimensionless, it must be a function of a
variable that is also dimensionless (similar to the way that $\sin(\theta)$ is
a function of $\theta$). Let us call this dimensionless variable $\xi$. With
$x$ and $\ell(t)$ to work with, only the ratio is dimensionless, so we deduce
that $\xi=x/\ell(t)$. Thus, we expect

$C(x,t)=\frac{1}{\ell(t)}F\left(\frac{x}{\ell(t)}\right).$ (6)

Let us now see if this form is consistent with the data. First, we note that
it guarantees that the total number of cells,
$N=\int_{-\infty}^{\infty}\\!\\!dx\leavevmode\nobreak\ C(x,t)$, does not
change with time because

$N=\int_{-\infty}^{\infty}\\!\\!dx\leavevmode\nobreak\
C(x,t)=\int_{\infty}^{\infty}dx\frac{1}{\ell(t)}F\left(\frac{x}{\ell(t)}\right)=\int_{-\infty}^{\infty}\\!d\xi
F(\xi),$ (7)

and $\int_{-\infty}^{\infty}\\!d\xi F(\xi)$ is a number that does not depend
on time (just like $\int_{0}^{\pi}d\theta\sin(\theta)$ is a number). Given
equation (6), the peak concentration $C(0,t)$ is just $F(0)/\ell(t)$, where
$F(0)$ is again just a number. With the scaling in (5) we deduce that
$C(0,t)\sim 1/\sqrt{t}$. A replotting of the data in fig2c on a log-log scale
shows that this is true (fig5a).

A significant prediction of the analysis leading to (6) is that the data at
different times should collapse when plotted as $C(x,t)/C(0,t)$ versus
$x/\ell(t)$, for this ratio is just $F(\xi)/F(0)$. (Dividing $C(x,t)$ by
$C(0,t)$ means that we rescale the heights of the various curves; and dividing
$x$ by $\ell(t)$ means that we allow for expansion of the initial
concentration of cells). If this holds, then it implies that $\ell(t)$ is the
only characteristic length in the system. A test of this is shown in fig5b,
where we see a good collapse of the data to a universal curve.

Figure 5: Rescaling the data. (a) The peak amplitude,$C(0,t)$, from fig2c
plotted as a function time, $t$, on a log-log scale; the dashed magenta line
has a slope of $-1/2$, which shows that $C(0,t)\sim t^{-1/2}$. (b) When the
data in fig2a are rescaled (see main text) and replotted, they collapse to a
universal curve; the dashed magenta curve is the function $\exp(-\xi^{2}/2)$

It is natural to seek a differential equation that is consistent with the
scaling $x^{2}\sim t$ and would provide a quantitative prediction of the
function $F$. First we consider if inertia is relevant in this system. We know
from fluid dynamics that inertia is irrelevant when the Reynolds number
$Re=UL/\nu$ is much less than unity: $U$ is the typical speed of a particle,
$L$ is the typical length of a particle, and $\nu=\eta/\rho$ is the kinematic
viscosity (which is defined as $\nu=\eta/\rho$, where $\eta$ is the fluid
viscosity and $\rho$ is the fluid density). For Chlamydomonas swimming in
water ($U\sim 10^{-2}$ cm/s, $L\sim 10^{-3}$ cm, and $\nu=10^{-2}$ cm2/s), we
have $Re\sim 10^{-3}$ and inertia is indeed negligible.

The differential equation we seek will have derivatives both in time and in
space. In the absence of inertia, we expect that the equation for $C(x,t)$
should only involve first-order derivatives in time (as second derivatives
would imply inertia and accelerations). With the scaling $x^{2}\sim t$ we
expect two space derivatives for one time derivative, so a consistent equation
would be

$\frac{\partial C}{\partial t}=D\frac{\partial^{2}C}{\partial^{2}x},$ (8)

where the parameter $D$ should be proportional to the empirical
${\mathscr{D}}$ obtained from fig2b.

To find a solution of (8) in the form of (6), we use $D$ to construct a length
$l=\sqrt{Dt}$ and find (see Mathematical Details) the normalized distribution

$C(x,t)=\frac{1}{\sqrt{4\pi Dt}}\exp{\left(-\frac{x^{2}}{4Dt}\right)}.$ (9)

Given this distribution, we compute the variance as

$\langle x^{2}\rangle=\int_{-\infty}^{\infty}x^{2}C(x,t)=2Dt.$ (10)

Comparing with our empirical observation (5), we deduce the relationship
${\mathscr{D}}=2D$ (the promised factor of two!) and therefore that the
dimensionless function is $F(\xi)=(2\pi)^{-1/2}\exp(-\xi^{2}/2)$. The ratio
$F(\xi)/F(0)=\exp(-\xi^{2}/2)$ is shown as the dashed line in fig5b, in good
agreement with the data.

Taken together, the experimental observations in fig2 and the phenomenological
analysis above, confirmed in fig5, suggest that the diffusion equation in (8)
provides a sound description of the spreading of cells that execute random
motions. It indicates that different organisms, with different diffusion
constants, obey the same fundamental scaling laws, insensitive to the details
of the underlying random motions. Note that at this level of analysis we do
not have a microscopic interpretation of the diffusion constant in terms of
the fluid viscosity and aspects of cell motility; it is simply a
phenomenological parameter that can be used to characterize a given
microorganism. On the other hand, if we knew from microscopical observations
that an organism’s motion consists of straight segments interrupted by random
reorientations, as in the case of Chlamydomonas and indeed E. coli (Berg,
1993), then by dimensional analysis (again) we could deduce
$D\sim\Delta^{2}/\tau\sim U^{2}\tau$ in terms of the run length $\Delta$,
speed $U$, and time between turns $\tau$.

##  3 Discussion

I have presented two ways of interleaving data and theory in a Results section
as a way of indicating how quantitative principles can be used to derive new
insight into phenomena. In one, a microscopic model led directly to the
diffusion equation, whose structure led to the ’rediscovery’ of Fick’s law,
which was confirmed from the data. In the second, the principles of
dimensional analysis and some phenomenological reasoning led us to postulate a
’new’ diffusion equation as a concise encoding of the experimental
observations. Each of these approaches used nothing more than basic algebraic
manipulations and elementary differential equations.

Returning to the referees who spoke of inferences drawn directly from the
data, I would ask: "What language does the data speak?" The answer would
appear to depend on one’s background. The inferences I drew from Lamarr’s data
were based on experience with understanding continuum and nonequilibrium
phenomena, subjects which are less common in the undergraduate physics
curriculum than one would hope, and very seldomly found in biology curricula.
So, I would indeed advocate a more holistic education for both biologists and
physicists (Goldstein et al., 2005).

It might be argued that the particular example I presented here is unusual,
but in fact these very same considerations (dimensional analysis, scaling
collapse of data, etc.) are to be found in many other places in biophysics.
Excellent examples are work on metabolic scaling laws (West et al., 1997) and
on stem cell replacement dynamics (Lopez-Garcia et al., 2010).

More importantly, I am not trying to emphasize any particular method in the
physicist’s toolbox, but rather a mindset that is about model-building and
testing as part of the results presented to the reader. This mindset is
particularly relevant when the theory is formulated first and the experiment
is undertaken to test it. But even when the experiment comes first there may
be a need to use theory as a sanity check on one’s observations (Meister,
2016). This also brings us to the delicate issue of the extent to which
research should actually be ’hypothesis driven’, as discussed provocatively by
Milner (2018): I will leave that Pandora’s box closed for the moment.

Finally, one could argue that the diffusion equation is ’just a model’ or
’just a theory’ and should, therefore, not be considered as a Result because,
unlike the data, it could be shown to be incorrect. With my experimentalist
hat on, I find that argument weak: almost every experiment has potentially
confounding aspects, and despite our best efforts to control them, these
effects can produce spurious results. After all, how many hundreds or
thousands of papers must have been written about stomach ulcers before
Marshall and Warren (1984) discovered that H. pylori was so often the culprit?
So, while it is certainly the case that many of the models discussed in
biology papers do not have the status of fundamental laws, I think that it is
contrary to the scientific method to view the fact that they may be superseded
as a weakness. If theories are crafted the right way they have utility even if
proven wrong, sometimes especially if proven wrong!

This essay has touched on two tensions - between theory and experiment, and
between the cultures of physics and biology. The differences between the
cultures have implications not only for how data is interpreted, but also for
what qualifies as "interesting" and who gets to frame the questions: an
enlightening debate on this issue was aired more than 20 years ago by Adrian
Parsegian and Robert Austin (Parsegian, 1997; Huebner et al., 1997). For
example, it might be argued that biologists may not really be interested in
the fact that a new equation has been derived that provides an approximate
description of a given system, and this could be a reason not to publish a
theoretical work in a biology journal. The example I provide here shows how
this need not be an empty exercise, but can lead to testable, mechanistic
predictions such as the relationship between flux and concentration gradient
(Fick’s Law, rediscovered). One need only consult the seminal work of Turing
(1952) on biological pattern formation or of Hodgkin and Huxley (1952) on
action potentials to see the importance of having a mathematical encoding of
diffusion to study its mechanistic implications. Likewise, a physics-oriented
experimental paper, even one that deals with living organisms, may also not be
seen as interesting to biologists because the questions appear unfamiliar. For
truly interdisciplinary journals, easing this tension is perhaps the greatest
challenge.

##  4 Methods

###  4.1 Generating the data

Full disclosure - rather than do the experiments, I numerically solved the
Langevin equation $dx/dt=\eta(t)$ for the time evolution of the position
$x(t)$ for a single alga undergoing random motion, where $\eta(t)$ is a random
variable with zero mean and temporal correlation function
$\langle\eta(t)\eta(t^{\prime})\rangle=2D\delta(t-t^{\prime})$. In the results
described here, I set $D=0.1$ mm2/s, approximately that of Chlamydomonas
(Polin et al., 2009). The equation was integrated forward a time increment
$\delta t$ from time index $i$ to $i+1$ using the discrete representation
$x_{i+1}=x_{i}+\sqrt{2D\delta t}\eta_{i}$, where $\eta_{i}$ is a normally
distributed random variable. The data represent averages over $30,000$
realizations.

###  4.2 Mathematical details

To obtain the normalized concentration profile (9) we simply substitute the
latter into the diffusion equation (8), with $\chi=x/\sqrt{Dt}$. We obtain

$\frac{d^{2}F}{d\chi^{2}}+\frac{1}{2}\left(F+\chi\frac{dF}{d\chi}\right)=0.$
(11)

Integrating (11) once and imposing the boundary condition that $F\to 0$ as
$\chi\to\infty$ we obtain $dF/d\chi+(1/2)\chi F=0$, which integrates to

$F(\xi)=A\exp(-\chi^{2}/4).$ (12)

Normalizing the associated concentration profile and re-expressing the result
in terms of the original variables yields the result (9).

##  5 Acknowledgments

I am grateful to Eric Lauga and Kyriacos Leptos for discussions, to Markus
Meister, Philip Nelson, Thomas Powers, Howard Stone, Kirsty Wan, Ned Wingreen,
and Francis Woodhouse for reviewing drafts of this essay, This work was
supported in part by an Investigator Award from the Wellcome Trust and an
Established Career Fellowship from the EPSRC. Apologies to Betteridge and
Hinchliffe for violating their laws of article titles.

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