Biophysical Applicationsof the Physics of DiffusionSima SetayeshgarDepartment of Physics,Indiana University9 July, 2007 Boulder Summer School, Applications of Diffusion 1Biochemical Signaling and thePhysics of Diffusion9 July, 2007 Boulder Summer School, Applications of Diffusion 2Measurement of concentration of adiffusing signaling molecule by areceptor is a generic task.In many cases, signaling moleculesare present in small copy numbers,making fluctuations in their num-bers significant.TFCheY-PCheYRNAPHow reliably can biochemical reactions be carried outgiven inherent fluctuations in numbersof crucial molecules?9 July, 2007 Boulder Summer School, Applications of Diffusion 3What is the physical limit in measuring the concentration ofsignaling molecules by biological receptors?First addressed by:H. C. Berg & E. M. Purcell, ``Physics of Chemoreception'', Biophysical Journal (1977)• Measurement of chemoattractant by single celled organism limited by statisticalfluctuations• Least fractional error attainable set by physics of diffusion• E. coli chemotaxis machinery near optimalOutline:• Revisit BP result within general framework of statistical mechanics• Generalize to cooperatively interacting receptor cluster• Compare with recent experiments9 July, 2007 Boulder Summer School, Applications of Diffusion 4E. coli as a Model OrganismWorkhorse of molecular biology;most studied cell in all of science:--- small genome (∼ 4300 genes),--- normal lack of pathogenicity--- ease of growth in the lab• Basis of recent developments inbiotechnology and genetic engineering,including living factory for producinghuman medicines• Basis for understanding of fundamentalcellular processes:--- cellular sensory systems,--- regulation of gene expression,--- cell division, etc.• Size:--- 0.5 µm in diameter,--- 1.5 µm in length• Cell cycle:--- ∼ 1 hour9 July, 2007 Boulder Summer School, Applications of Diffusion 5Sensory Mechanisms in Single Cells:Chemotaxis in E. coli3D tracking microscope imageof a single cell's motionFrom H. C. Berg, Physics Today, 2000.Biased random walk of runspunctuated by tumbles:• Temporal measurement of externalconcentration• Response:Modulation of mean runtime• Physical constants:τT∼ 0.1 sτR∼ 1 s (in uniform environment)v ∼ 20 µm/sfrom E. O. Budrene and H. C. Berg,Nature (1995)9 July, 2007 Boulder Summer School, Applications of Diffusion 6E. coli Chemotaxis Signaling Networkrestores tumbling.Methylationsuppresses tumbling.Attractant(external signal)Attractant/Repellent P(Fast)(internal messenger)(Fast)CheYCHCH33Receptor + CheA + CheWMotorCheR(Slow)CheZ(Fast)(Fast)(feedback)(methyl transferase)(Slow)CheB(methyl esterase)Example of• Extracellular signaling: cell measures external attractant/repellent concentration• Intracelluar signaling: motor measures [CheY-P]9 July, 2007 Boulder Summer School, Applications of Diffusion 7Berg-Purcell Results Revisited9 July, 2007 Boulder Summer School, Applications of Diffusion 8``Perfect'' Device• Single measurement of number of substrate molecules:N = N ± δN1, N ∼ ca3, δN1∼pN• Diffusion time: τD∼ a2/D• Number of independent measurements in time τ :NM∼ τ /τD, δNM∼qN/NMδc/c ∼ 1/√caDτ9 July, 2007 Boulder Summer School, Applications of Diffusion 9Questions• How does this argument based on counting molecules in a volumeapply to cells and receptors that count molecules on their surface?• What about the details of the biochemical kinetics that govern theinteraction of ligand with its receptor?• How to generalize from a single receptor to a cluster of receptors?What about interactions between the receptors, or between thereceptors and internal states of the cluster?9 July, 2007 Boulder Summer School, Applications of Diffusion 10Multiple Surface AbsorbersMultiple AbsorbersFor Nrdiscrete absorbers of size b,• By analogy with electrostatics:I/Imaxsphere= Nrb / (Nrb + πa)• Also from:1 − Pesc= Nrb / (Nrb + 4a)Pesc: probability that substratemolecule at r = a + bsurvives all subsequentcontacts and escapes to ∞Beat down diffusion noise by 1/√Ntot:δcrmsc∼1√DcτπNrb+1a1/29 July, 2007 Boulder Summer School, Applications of Diffusion 11Integration Time?• Cell swimming along x axis wan-ders off course due to rotationaldiffusion about y and z axes:Dr= kBT /γr, γr= 8πηa3• Bacterium wanders off byθ2= 4Drτ in time τ .• Integration time must be shorterthan the time for rotationaldiffusion to disorient the cell!Question:• How does the E. coli's mean runtime comparewith its characteristic timescale for rotational diffusion?9 July, 2007 Boulder Summer School, Applications of Diffusion 12Statistical MechanicsTreatmentW. Bialek and S. Setayeshgar, Proc. Nat'l. Acad. Sci. (USA) 102, 10040 (2005).9 July, 2007 Boulder Summer School, Applications of Diffusion 13Linear ResponseLinear response of receptor occupancy to conjugate `force':d δndt= −(k+c + k−) δn + k−(1 − n)[δc + c β δF]d δcdt= D∇2δc − δ(~x − ~x0)d δndtGeneralized susceptibility:˜α(ω) =δ˜n(ω)δ˜F (ω)=k+c(1 − n)kBT1−iω[1 + Σ(ω)] + (k+c + k−)Σ(ω) = k+(1 − n)ZΛ0d3k(2π)31−iω + Dk2, Λ ∼ π/aFluctuation-dissipation theorem:SF(ω) = −2kBTωIm"δ˜F (ω)δ˜n(ω)#Effective spectral density of noise in measuring c, in terms of`noise force' spectrum:Seffc(ω) =ckBT2SF(ω)9 July, 2007 Boulder Summer School, Applications of Diffusion 14Linear ResponseLinear response of receptor occupancy to conjugate ``force'':d δndt= −(k+c + k−) δn + k+(1 − n) [δc + cβ δF ]| {z }δF · cβAnalogous to Brownian motion in a harmonic potential:M¨δX + γ˙δX + k δX = δf(t)(ignoring the inertial term, M¨δX)9 July, 2007 Boulder Summer School, Applications of Diffusion 15Mechanical/Chemical SystemsPhysical quantity Mass-spring system Chemical systemCoordinate Displacement Receptor occupancy:δn = n − nConjugate force f(t) Free energy change:δF = kBT (δk+/k+− δk−/k−)`Spring' constant k kBT /[n(1 − n)]`Damping' constant γ kBT /(k−n)• Fluctuation-dissipation theoremδf(t) · δf(t0)= 2kBT γ δ(t − t0)• More generally,δX(t) =Zdt0α(t − t0) δf (t0)SX(ω) =2kBTωIm[˜α(ω)], Sf(ω) = −2kBTωIm1˜α(ω)9 July, 2007 Boulder Summer School, Applications of Diffusion 16Connection with Berg-Purcell ResultAccuracy of a measurement which integrates for a time τ  τc:δcrms≈rSeffc(ω = 0) ·1τTwo contributions to