Are theoretical results 'Results'?
MManuscript submitted to eLife
Are theoretical results ’Results’?
Raymond E. Goldstein * *For correspondence: [email protected] Department of Applied Mathematics and Theoretical Physics, Centre for MathematicalSciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
Abstract
Yes.
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 combinedexperimental biology and physical calculations in our paper: "The data should be described andthe inferences drawn, and the modelling relegated to its proper place as quantitative verification ofthe 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 authorsshould let the data speak for itself, and postpone heavier theoretical analysis for later, perhapsin the Discussion." Many of my colleagues have experienced the same reaction to papers mixingtheory 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 experiencewith some in the biology community, namely that they see the significance of theory very differentlyfrom 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/Discussiondichotomy 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: harmonicoscillators, 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-definedset of assumptions leads to precise outcomes, and where the dependence of the outcomes on allof the parameters can be interpreted. This approach allows us to estimate what is important andwhat is not in any setting. Models also help us to think about problems: "If this is the underlyingphysics, then A should vary with B quadratically...", or "under these assumptions, the data shouldcollapse like this..." or, when we spot something is not quite right, "here I argue that these claimsare in conflict with basic laws of physics" (
Meister, 2016 ).The role of theory is also intimately connected with predictions . While I know biologists whowould say “who cares about a prediction in the absence of experiment?”, physicists are broughtup to celebrate them - they are the stuff of legend, from Dirac’s prediction of antiparticles andEinstein’s prediction of the bending of starlight, to the work by many that predicted the Higgsparticle. We view predictions as motivations for experiment and as a means to move the disciplineforward. 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 todesign experiments with theory in mind, as emphasized by
Bialek ( ), who has described manyhistorical examples of the role theory has played in biology, from Rayleigh’s work on hearing toWatson and Crick. a r X i v : . [ q - b i o . O T ] J u l anuscript submitted to eLifeMy purpose here is to push back against the view that theory is not a ’Result’. I argue forthe unabashed inclusion of mathematical formulations and pedagogy within the body of paperspublished in eLife and other primarily biological journals. By interleaving the experimental andtheoretical results it is possible to tell a story, and I firmly believe this makes for much moreinteresting and readable papers. It is also faithful to the scientific method, in which one goes backand forth with experiment and hypothesis.Readers may be interested to learn that biological information, background and results arenow routinely included in papers published in physics journals, although this has not always beenthe 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 deskand asked: "Why would any physicist study something as ridiculous as that?" But by now manyphysicists do exactly that, and many physics journals are full of discussions of cAMP signaling, spiralwaves, and chemotaxis (
Goldstein, 1996 ; Rappel et al., 1999 ; Gholami et al., 2015 ). If we really takeinterdisciplinary research seriously then I assert there has to be a prominent place for theory withinbiology papers, both as Results in papers that combine experiment and theory, and as Results intheory papers.This is nothing new. If you have not already done so, I highly recommend reading the celebratedpaper by
Hodgkin and Huxley ( ) to see experiments and theory interleaved. Theory is notrelegated to the discussion, or worse, to supplementary material, but instead is incorporated intothe 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 wasappropriate a century ago, why must details of mathematical models now be relegated to the backof papers (see, for example,
Paulick et al. ( ), Ferreira et al. ( ), and Streichan et al. ( ))?Many readers will appreciate that the issue I am raising about quantitative descriptions ofliving systems is closely associated with the tension that exists between the stereotypes of thebiologist, who wants to incorporate all the complexity of a particular system, and the physicistwho 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 thisdevelopment 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 howphysicists 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 readerthrough an example of how (at least some) physicists might go about describing a well-knownphenomenon that shows up everywhere in biology - from the functioning of cellular receptorsto bacterial chemotaxis, the propagation of action potentials, and fluorescence recovery afterphotobleaching (FRAP) - namely, diffusion. Employing poetic license, I imagine that we are at apoint in time when the diffusion equation itself was not known, nor was Fick’s Law, so both theexperimental observations and theoretical analysis presented below are new and worthy of beingdescribed as Results.I compose two versions of a Results section to indicate various ways of presenting the dataand theory interleaved in a compact presentation that (I hope) is widely understandable by thecommunity. The first version involves a ’microscopic’ model that is a caricature of the biologicalsystem, 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 probablymore challenging - involves the use of ’dimensional analysis’, one of the most powerful methodsof analyzing natural phenomena. Here, relationships between various quantities are deduced byexamining the units in which they are measured (mass, length, time, charge, etc.). Introduced longago, particularly in the work of
Maxwell ( ), this technique can often lead to exact answers toproblems, up to the proverbial ’factors of two’. anuscript submitted to eLife Figure 1.
Experimental setup to study diffusion in the green alga
Chlamydomonas . (a) A light sheet is used togather the algae, which are swimming in a petri dish, into a narrow strip of cells along the 𝑦 -axis. (b) After thelight is turned off, the cells swim randomly and spread out. The concentration profile, 𝐶 ( 𝑥, 𝑡 ) , is then measuredalong a thin strip parallel to the 𝑥 -axis; 𝑡 is time. A discovery
Allow me to introduce our fictitious Professor Lamarr, who has been investigating how the single-cellgreen alga
Chlamydomonas move in response to light. She has discovered that if a narrow sheet oflight is directed into an algal suspension in a petri dish (
Figure 1a ), the algae swim into the beamand form a concentrated line of cells. When the light is turned off and there is no more phototacticcue, the cells resume a random swimming motion described previously (
Polin et al., 2009 ), in whichevery s or so their roughly linear motion is interrupted by a turn: the angle of this turn falls withina distribution that has a mean of ∼ 90 degrees. These random turns lead the population to spreadout over time ( Figure 1b ). See
Methods for experimental details.Lamarr measures the normalized concentration profiles, 𝐶 ( 𝑥, 𝑡 ) , in a thin strip that is perpendic-ular to the initial line of cells, obtaining the data shown in Figure 2a . The sharply-peaked profile atearly times gradually spreads out until the Petri dish is uniformly filled with cells. She measuredthe variance ⟨ 𝑥 ⟩ of the concentration profile, and found the linear relation ⟨ 𝑥 ⟩ = 𝒟 𝑡 , with 𝒟 = 0 . mm /s ( Figure 2b ). Finally, the peak height 𝐶 (0 , 𝑡 ) decays smoothly with time ( Figure 2c ). By sys-tematic experimentation, she found that the basic results were insensitive to the precise size ofthe initial gathering, and that various swimming mutants of
Chlamydomonas displayed the samebehavior, albeit with different values of 𝒟 . (a) (b)(c) Figure 2.
Experimental results on diffusion in a population of the green alga
Chlamydomonas . (a) Concentrationprofiles, 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, ⟨ 𝑥 ⟩ , of the data shown in (a) as a function of time; the dashedmagenta 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.3 of 9 anuscript submitted to eLife 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 individualcells and deduce from it a population-level description with which to analyze the data. In thesimplest picture, we assume that cells move only to the left and right along the 𝑥 -axis, and thecells are constrained to sit on a discrete set of points, at positions 𝑥 𝑚 = 𝑚 Δ , where 𝑚 = 1 , , , … ( Figure 3a ). Likewise, we assume time is discrete, so at each time 𝑡 𝑛 = 𝑛𝜏 , 𝑛 = 1 , , , … , a cell moveswith probability to the left or right, as indicated by the arrows in Figure 3a . Figure 3.
A random walk in one dimension. (a) A cell at site 𝑚 moves with probability 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 Δ 𝑥 at time 𝑛 Δ 𝑡 we observe ( Figure 3b ) that cells that appear at point 𝑚 at time 𝑛 + 1 arrived there bymoving to the right from point 𝑚 − 1 or by moving to the left from point 𝑚 + 1 at the previous timestep (each with probability ). Thus we can deduce that 𝐶 𝑛 +1 ( 𝑚 ) = 12 𝐶 𝑛 ( 𝑚 + 1) + 12 𝐶 𝑛 ( 𝑚 − 1) . (1)We now imagine that the probabilities are varying sufficiently slowly in space and time that wecan use the following Taylor expansions: 𝐶 𝑛 +1 ( 𝑚 ) ≃ 𝐶 𝑛 ( 𝑚 ) + 𝜏 ( 𝜕𝐶 𝑛 ( 𝑚 )∕ 𝜕𝑡 ) + ⋯ ; and 𝐶 𝑛 ( 𝑚 ± 1) ≃ 𝐶 𝑛 ( 𝑚 ) ± Δ( 𝜕𝐶 𝑛 ( 𝑚 )∕ 𝜕𝑥 ) + (Δ ∕2)( 𝜕 𝐶 𝑛 ( 𝑚 )∕ 𝜕𝑥 ) + ⋯ . Collecting terms, we deduce that the ’continuumlimit’ for this one-dimensional random walk is 𝜕𝐶𝜕𝑡 = 𝐷 𝜕 𝐶𝜕𝑥 , with 𝐷 = Δ 𝜏 . (2)We term this the ‘diffusion equation’, where the diffusion constant 𝐷 has units of length /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 Δ as the typical distance a cell travelsbetween sharp turns, and 𝜏 as the time between such turns. If 𝑈 is the swimming speed betweenturns, then Δ ∼ 𝑈𝜏 , so we can write 𝐷 = 𝑈 𝜏 ∕2 . From tracking studies of Chlamydomonas , we knowthat 𝑈 ∼ 0 . mm/s, and 𝜏 ∼ 10 s, and therefore Δ ∼ 1 mm and 𝐷 ∼ 0 . mm /s.If we rewrite the diffusion equation (2) as 𝜕𝐶 ∕ 𝜕𝑡 = −( 𝜕 ∕ 𝜕𝑥 )(− 𝐷𝜕𝐶 ∕ 𝜕𝑥 ) then it can be written as 𝜕𝐶𝜕𝑡 = − 𝜕𝐽𝜕𝑥 , where 𝐽 = − 𝐷 𝜕𝐶𝜕𝑥 , (3)where we identify the flux 𝐽 as the number of cells passing through a given point 𝑥 per unittime. This relationship implies that cells pass from regions of high concentration to low at a rateproportional the gradient of concentration. This ’flux form’ of the diffusion equation guaranteesthat the total number of cells, 𝑁 = ∫ ∞−∞ 𝑑𝑥𝐶 ( 𝑥, 𝑡 ) , remains constant over time, since 𝑑𝑁𝑑𝑡 = ∫ ∞−∞ 𝑑𝑥 𝜕𝐶 ( 𝑥, 𝑡 ) 𝜕𝑡 = − ∫ ∞−∞ 𝑑𝑥 𝜕𝐽𝜕𝑥 = 𝐽 (−∞) − 𝐽 (+∞) . (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𝜕𝐶 ∕ 𝜕𝑥 can be tested experimentally. Lamarr recorded thedistributions of cells at the times indicated in Figure 2 and then again . s later. As shown in anuscript submitted to eLife Figure 4a for one pair, such measurements yield the flux, 𝐽 , and concentration gradient, 𝜕𝐶 ∕ 𝜕𝑥 each as functions of 𝑥 ( Figure 4b ), and we see that, apart from the overall scale, they are oppositelysigned, as predicted by (3). But we can now go one step further and plot 𝐽 at each point 𝑥 and time 𝑡 versus 𝜕𝐶 ∕ 𝜕𝑥 at those same 𝑥 and 𝑡 values. If the theory is correct, then every data set shouldcollapse on to a single straight line, and indeed this is the case ( Figure 4c ). According to the theoryabove, the slope of the line in
Figure 4c is the diffusion constant 𝐷 ; we obtain 𝐷 = 0 . mm /s, whichis consistent with the microscopic interpretation in terms of motility. (a) (b)(c) Figure 4.
Flux and the diffusion equation. (a) Concentration profiles, 𝐶 ( 𝑥, 𝑡 ) , at times 𝑡 = 3 and 𝑡 = 3 . s . (b)The flux of cells past a given point, 𝐽 (black; left axis), and the concentration gradient, 𝜕𝐶 ∕ 𝜕𝑥 (yellow; right axis),versus position, 𝑥 . (c) Flux, 𝐽 , versus concentration gradient, 𝜕𝐶 ∕ 𝜕𝑥 , for all the values of 𝑥 and 𝑡 shown in Figure 2a . The dashed magenta line has a slope 𝐷 = 0 . mm /s. 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 equationfor the time evolution of the algal concentration 𝐶 ( 𝑥, 𝑡 ) , which is measured in organisms per mm,hence units of 1/length. The variance ⟨ 𝑥 ⟩ has, of course, units of length squared, so we can definea characteristic, time-dependent length 𝓁 ( 𝑡 ) = √⟨ 𝑥 ⟩ . From the fit to the data in Figure 2b we inferthat the width of 𝐶 ( 𝑥, 𝑡 ) grows as 𝓁 ( 𝑡 ) ∼ √ 𝒟 𝑡. (5)A very natural question is whether 𝓁 ( 𝑡 ) is the only intrinsic length scale that can be extractedfrom the data. As 𝐶 ( 𝑥, 𝑡 ) has units of number/length we can, without loss of generality, write 𝐶 ( 𝑥, 𝑡 ) = 𝓁 ( 𝑡 ) −1 𝐹 ( 𝑥, 𝑡 ) for some unknown function 𝐹 that is itself dimensionless. And since 𝐹 isdimensionless, it must be a function of a variable that is also dimensionless (similar to the way that sin( 𝜃 ) is a function of 𝜃 ). Let us call this dimensionless variable 𝜉 . With 𝑥 and 𝓁 ( 𝑡 ) to work with, onlythe ratio is dimensionless, so we deduce that 𝜉 = 𝑥 ∕ 𝓁 ( 𝑡 ) . Thus, we expect 𝐶 ( 𝑥, 𝑡 ) = 1 𝓁 ( 𝑡 ) 𝐹 ( 𝑥 𝓁 ( 𝑡 ) ) . (6)Let us now see if this form is consistent with the data. First, we note that it guarantees that thetotal number of cells, 𝑁 = ∫ ∞−∞ 𝑑𝑥 𝐶 ( 𝑥, 𝑡 ) , does not change with time because 𝑁 = ∫ ∞−∞ 𝑑𝑥 𝐶 ( 𝑥, 𝑡 ) = ∫ ∞∞ 𝑑𝑥 𝓁 ( 𝑡 ) 𝐹 ( 𝑥 𝓁 ( 𝑡 ) ) = ∫ ∞−∞ 𝑑𝜉𝐹 ( 𝜉 ) , (7) anuscript submitted to eLifeand ∫ ∞−∞ 𝑑𝜉𝐹 ( 𝜉 ) is a number that does not depend on time (just like ∫ 𝜋 𝑑𝜃 sin( 𝜃 ) is a number). Givenequation (6), the peak concentration 𝐶 (0 , 𝑡 ) is just 𝐹 (0)∕ 𝓁 ( 𝑡 ) , where 𝐹 (0) is again just a number. Withthe scaling in (5) we deduce that 𝐶 (0 , 𝑡 ) ∼ 1∕ √ 𝑡 . A replotting of the data in Figure 2c on a log-logscale shows that this is true (
Figure 5a ).A significant prediction of the analysis leading to (6) is that the data at different times shouldcollapse when plotted as 𝐶 ( 𝑥, 𝑡 )∕ 𝐶 (0 , 𝑡 ) versus 𝑥 ∕ 𝓁 ( 𝑡 ) , for this ratio is just 𝐹 ( 𝜉 )∕ 𝐹 (0) . (Dividing 𝐶 ( 𝑥, 𝑡 ) by 𝐶 (0 , 𝑡 ) means that we rescale the heights of the various curves; and dividing 𝑥 by 𝓁 ( 𝑡 ) means thatwe allow for expansion of the initial concentration of cells). If this holds, then it implies that 𝓁 ( 𝑡 ) isthe only characteristic length in the system. A test of this is shown in Figure 5b , where we see agood collapse of the data to a universal curve. (a) (b)
Figure 5.
Rescaling the data. (a) The peak amplitude, 𝐶 (0 , 𝑡 ) , from Figure 2c plotted as a function time, 𝑡 , on alog-log scale; the dashed magenta line has a slope of −1∕2 , which shows that 𝐶 (0 , 𝑡 ) ∼ 𝑡 −1∕2 . (b) When the data in Figure 2a are rescaled (see main text) and replotted, they collapse to a universal curve; the dashed magentacurve is the function exp(− 𝜉 ∕2) It is natural to seek a differential equation that is consistent with the scaling 𝑥 ∼ 𝑡 and wouldprovide a quantitative prediction of the function 𝐹 . First we consider if inertia is relevant inthis system. We know from fluid dynamics that inertia is irrelevant when the Reynolds number 𝑅𝑒 = 𝑈𝐿 ∕ 𝜈 is much less than unity: 𝑈 is the typical speed of a particle, 𝐿 is the typical length ofa particle, and 𝜈 = 𝜂 ∕ 𝜌 is the kinematic viscosity (which is defined as 𝜈 = 𝜂 ∕ 𝜌 , where 𝜂 is the fluidviscosity and 𝜌 is the fluid density). For Chlamydomonas swimming in water ( 𝑈 ∼ 10 −2 cm/s, 𝐿 ∼ 10 −3 cm, and 𝜈 = 10 −2 cm /s), we have 𝑅𝑒 ∼ 10 −3 and inertia is indeed negligible.The differential equation we seek will have derivatives both in time and in space. In the absenceof 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 ∼ 𝑡 we expect twospace derivatives for one time derivative, so a consistent equation would be 𝜕𝐶𝜕𝑡 = 𝐷 𝜕 𝐶𝜕 𝑥 , (8)where the parameter 𝐷 should be proportional to the empirical 𝒟 obtained from Figure 2b .To find a solution of (8) in the form of (6), we use 𝐷 to construct a length 𝑙 = √ 𝐷𝑡 and find (see Mathematical Details ) the normalized distribution 𝐶 ( 𝑥, 𝑡 ) = 1 √ 𝜋𝐷𝑡 exp ( − 𝑥 𝐷𝑡 ) . (9)Given this distribution, we compute the variance as ⟨ 𝑥 ⟩ = ∫ ∞−∞ 𝑥 𝐶 ( 𝑥, 𝑡 ) = 2 𝐷𝑡. (10)Comparing with our empirical observation (5), we deduce the relationship 𝒟 = 2 𝐷 (the promisedfactor of two!) and therefore that the dimensionless function is 𝐹 ( 𝜉 ) = (2 𝜋 ) −1∕2 exp(− 𝜉 ∕2) . The ratio 𝐹 ( 𝜉 )∕ 𝐹 (0) = exp(− 𝜉 ∕2) is shown as the dashed line in Figure 5b , in good agreement with the data. anuscript submitted to eLifeTaken together, the experimental observations in
Figure 2 and the phenomenological analysisabove, confirmed in
Figure 5 , suggest that the diffusion equation in (8) provides a sound descriptionof the spreading of cells that execute random motions. It indicates that different organisms, withdifferent diffusion constants, obey the same fundamental scaling laws, insensitive to the details ofthe 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 issimply a phenomenological parameter that can be used to characterize a given microorganism. Onthe other hand, if we knew from microscopical observations that an organism’s motion consistsof straight segments interrupted by random reorientations, as in the case of
Chlamydomonas andindeed
E. coli ( Berg, 1993 ), then by dimensional analysis (again) we could deduce 𝐷 ∼ Δ ∕ 𝜏 ∼ 𝑈 𝜏 interms of the run length Δ , speed 𝑈 , and time between turns 𝜏 . Discussion
I have presented two ways of interleaving data and theory in a Results section as a way of indicatinghow quantitative principles can be used to derive new insight into phenomena. In one, a microscopicmodel 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 somephenomenological reasoning led us to postulate a ’new’ diffusion equation as a concise encoding ofthe experimental observations. Each of these approaches used nothing more than basic algebraicmanipulations 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 continuumand nonequilibrium phenomena, subjects which are less common in the undergraduate physicscurriculum than one would hope, and very seldomly found in biology curricula. So, I would indeedadvocate 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 thesevery same considerations (dimensional analysis, scaling collapse of data, etc.) are to be found inmany other places in biophysics. Excellent examples are work on metabolic scaling laws (
Westet 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 presentedto the reader. This mindset is particularly relevant when the theory is formulated first and theexperiment is undertaken to test it. But even when the experiment comes first there may be a needto use theory as a sanity check on one’s observations (
Meister, 2016 ). This also brings us to thedelicate issue of the extent to which research should actually be ’hypothesis driven’, as discussedprovocatively by
Milner ( ): 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 potentiallyconfounding aspects, and despite our best efforts to control them, these effects can producespurious results. After all, how many hundreds or thousands of papers must have been writtenabout stomach ulcers before Marshall and Warren ( ) discovered that H. pylori was so often theculprit? So, while it is certainly the case that many of the models discussed in biology papers do nothave the status of fundamental laws, I think that it is contrary to the scientific method to view thefact that they may be superseded as a weakness. If theories are crafted the right way they haveutility even if proven wrong, sometimes especially if proven wrong!This essay has touched on two tensions - between theory and experiment, and between thecultures of physics and biology. The differences between the cultures have implications not onlyfor how data is interpreted, but also for what qualifies as "interesting" and who gets to frame anuscript submitted to eLifethe questions: an enlightening debate on this issue was aired more than 20 years ago by AdrianParsegian and Robert Austin (
Parsegian, 1997 ; Huebner et al., 1997 ). For example, it might beargued that biologists may not really be interested in the fact that a new equation has been derivedthat provides an approximate description of a given system, and this could be a reason not topublish a theoretical work in a biology journal. The example I provide here shows how this neednot be an empty exercise, but can lead to testable, mechanistic predictions such as the relationshipbetween flux and concentration gradient (Fick’s Law, rediscovered). One need only consult theseminal work of
Turing ( ) on biological pattern formation or of Hodgkin and Huxley ( ) onaction potentials to see the importance of having a mathematical encoding of diffusion to study itsmechanistic implications. Likewise, a physics-oriented experimental paper, even one that deals withliving organisms, may also not be seen as interesting to biologists because the questions appearunfamiliar. For truly interdisciplinary journals, easing this tension is perhaps the greatest challenge. Methods
Generating the data
Full disclosure - rather than do the experiments, I numerically solved the Langevin equation 𝑑𝑥 ∕ 𝑑𝑡 = 𝜂 ( 𝑡 ) for the time evolution of the position 𝑥 ( 𝑡 ) for a single alga undergoing random motion, where 𝜂 ( 𝑡 ) is a random variable with zero mean and temporal correlation function ⟨ 𝜂 ( 𝑡 ) 𝜂 ( 𝑡 ′ ) ⟩ = 2 𝐷𝛿 ( 𝑡 − 𝑡 ′ ) .In the results described here, I set 𝐷 = 0 . mm /s, approximately that of Chlamydomonas ( Polinet al., 2009 ). The equation was integrated forward a time increment 𝛿𝑡 from time index 𝑖 to 𝑖 + 1 using the discrete representation 𝑥 𝑖 +1 = 𝑥 𝑖 + √ 𝐷𝛿𝑡𝜂 𝑖 , where 𝜂 𝑖 is a normally distributed randomvariable. The data represent averages over , realizations. Mathematical details
To obtain the normalized concentration profile (9) we simply substitute the latter into the diffusionequation (8), with 𝜒 = 𝑥 ∕ √ 𝐷𝑡 . We obtain 𝑑 𝐹𝑑𝜒 + 12 ( 𝐹 + 𝜒 𝑑𝐹𝑑𝜒 ) = 0 . (11)Integrating (11) once and imposing the boundary condition that 𝐹 → as 𝜒 → ∞ we obtain 𝑑𝐹 ∕ 𝑑𝜒 + (1∕2) 𝜒𝐹 = 0 , which integrates to 𝐹 ( 𝜉 ) = 𝐴 exp(− 𝜒 ∕4) . (12)Normalizing the associated concentration profile and re-expressing the result in terms of theoriginal variables yields the result (9). 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 reviewingdrafts of this essay, This work was supported in part by an Investigator Award from the WellcomeTrust and an Established Career Fellowship from the EPSRC. Apologies to Betteridge and Hinchliffefor violating their laws of article titles.
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