Introduction to the mathematical modeling ofmulti-scale phenomenaDiffusionMCB/MATH 303MCB/MATH 303 Multi-scale modeling - DiffusionBrownian motionImage (public domain) from J. B. Perrin,Mouvement Brownien et r´ealit´e mol´eculaire,Ann. de Chimie et de Physique 18, 5-114(1909).Brownian motion (named afterbotanist Robert Brown) refers tothe random motion of particlessuspended in a fluid as they are“pushed around” by the smallerfluid molecules.This phenomenon may be modeledin terms of a random walk.The diffusion MATLAB GUIillustrates that, on average, aparticle performing a random walkon the plane is, after time t, at adistance proportional to√t fromthe origin.MCB/MATH 303 Multi-scale modeling - DiffusionRandom walkTrajectories of three particlesperforming a 1000-step randomwalk on the plane.This immediately allows us tointroduce a dimensionlessquantity DT /L2, where T is acharacteristic time, L acharacteristic length, and D adiffusion coefficient.We recognize the samedimensionless combination thatappears in the heat equation,∂u∂t= D∂2u∂x2.One can in fact show that a random walk at the microscopic levelcan be described at the macroscopic level by a diffusion equation.MCB/MATH 303 Multi-scale modeling - DiffusionReaction-diffusion equationsThe dynamics of macroscopic dynamics of quantities thatboth diffuse and interact, such as chemicals in a reaction, istypically described in terms of a reaction-diffusion equation ofthe form∂u∂t= D∂2u∂x2+ F (u),where F is a reaction term.Reaction-diffusion systems often lead to the formation ofTuring patterns, such as hexagons or stripes.Such patterns, which have beenreproduced in chemical reactionexperiments, are also thoughtto be observed on animal coatsand fish skins.MCB/MATH 303 Multi-scale modeling - DiffusionMicroscopic versus macroscopic aspects of diffusionThe macroscopic description of a microscopic phenomenon istypically obtained by taking averages. As a consequence,individual aspects are lost and replaced by quantities such asdensities, which can be measured at the macroscopic level.Since a random walk at the microscopic level can be describedat the macroscopic level by a diffusion equation, the samereasoning may be applied to describe any motion that involvesa random walk, regardless of the sc ale at which it happens.In particular, flagellated bacteria such Bacillus subtilis swim ina series of runs and tumbles, which can be modele d by arandom walk.MCB/MATH 303 Multi-scale modeling - DiffusionMotion of B. subtilis on agarBacillus subtilis is a flagellated rod-like bacteriumMovie (made i n 2001 by then undergraduatestudent Cathy Ott in N. Mendelson’s lab) showingBacillus subtilis bacteria swimming on agar.Larger MovieLength: 2 to 3 µm.Diameter: ∼ 0.7 µm.Swimming speed:about 10 times itslength per se cond.It moves by asuccession of runs andtumbles.MCB/MATH 303 Multi-scale modeling - DiffusionColony forms for B. subtilisN.H. Mendelson, a nd B. Salhi, Patterns o f reporter gene expression in the phasediagram of B aci ll us subtilis colony forms, J. Bacteri ol. 178, 1980-1989 (1996).(Diameter of petri dishes: 6 cm)MCB/MATH 303 Multi-scale modeling - DiffusionReaction-diffusion modelsIt is therefore not surprising that branched bacterial colony shapesmay be captured by means of reaction-diffusion models.∂S∂t= DS∇2S − ηNSS : density of nutrientsN : density of bacteriaThese models often involve nonlinear diffusion,∂N∂t= ∇DNNk∇N+ NS − µNS. Kitsunezaki, J. Phys. Soc. Jpn. 66, 1544-1550 (1997)Possibly with a stochastic diffusion coefficient:∂N∂t= ∇DN(1 + σ)NS ∇N+ NSK. Kawasaki et al., J. Theor. B iol . 188, 177-185 (1997)Images from I. Gol ding et al., Physica A 260, 510-554 (1998).MCB/MATH 303 Multi-scale modeling - DiffusionForaging behaviorsIt is natural to push the analogy even further, to describe thedynamics of “objects” of even bigger size, such as animalsforaging for food.But of course, the scaling law L2∝ T is only valid if thedirections allowed for each step in the random walk areequally likely, and if steps are taken at regular intervals.In particular, if the random walker stops and stays put for awhile, the average distance L after time T will grow moreslowly than√T . This is called subdiffusion.Similarly, if the walker takes unusually large steps, L2willgrow faster than T and superdiffusion will be observed.The above ideas have been applied in ecology to describe theforaging behaviors of animals.MCB/MATH 303 Multi-scale modeling - DiffusionSummaryWe started with multi-scale asp ec ts of various physical orbiological systems.We introduced the concept of scales and discussed examplesof scale-free systems, such as fractals and self-similar systemsfound in nature.We then turned to a discussion of how ideas of scales app earin models that involve differential equations. This led us todimensional analysis and self-similar solutions of partialdifferential equations.We studied random walks and use d sc alings to associate themwith diffusion at the macroscopic level. This took us back topatterns via reaction-diffusion equations.Finally, we extended the concept of a random walk tophenomena occurring at the scales of a few microns (bacteria)and at our own scale (foraging animals).MCB/MATH 303 Multi-scale modeling - DiffusionMotion of B. subtilis on agarMovie by Cathy OttBackMCB/MATH 303 Multi-scale modeling -