Level crossing statistics in a biologically motivated model of a long dynamic protrusion: passage times, random and extreme excursions
LLength fluctuations of long cell protrusions:statistics of passage times, random and extremal excursions
Swayamshree Patra and Debashish Chowdhury ∗ Department of Physics, Indian Institute of Technology Kanpur, 208016, India
The functional consequences of convenient size of subcellular structures have been analysed in thepast and currently the questions regarding the underlying physical mechanisms for their genesis,the sensing of their size by the cell and maintenance of their convenient size in the steady state arereceiving much attention. Long cell protrusions, which are effectively one-dimensional, are highlydynamic subcellular structures and their length keep fluctuating about the mean length even in thethe steady state. For optimal functioning, particularly as sensors, the length of a specific protrusionmust not cross a narrow band bounded by two thresholds. However, fluctuations may drive thelength beyond these thresholds. We study generic stochastic models of length control of long cellprotrusions; the special cases of this model correspond to specific examples of different types of cellappendages. Exploiting the techniques of level crossings developed for random excursions of stochas-tic process, we have derived analytical expressions of passage times for hitting various thresholds,sojourn times of random excursions beyond the threshold and the extreme lengths attained duringthe lifetime of these protrusions. As a concrete example, we apply our general formulae to flagella(also called cilia) of eukaryotic cells thereby getting estimates of the typical length scales and timescales associated with the various aspects of random fluctuations. Since most of the earlier worksinvestigated only the mean length of cell protrusions, our study of the fluctuations opens a newhorizon in the field of subcellular size control.
I. INTRODUCTION
Cell is the structural and functional unit of life [1]. Forbiological function, size matters at all levels of biologicalorganization [2–4]. Even at the subcellular level, each ofthe structures in an eukaryotic cell has an optimal size [5–7]. Dynamic long cell protrusions, like eukaryotic flagellaand cilia, are effectively linear subcellular structures. Thetypical length of a particular type of protrusion showsnegligible variation across all members of a specific celltype in a given species under same physiological condi-tion. This observation suggests that the individual cellscontrol the length of these structures. What makes thesesystems attractive in the study of cellular mechanisms forsize control is that the the investigator has to deal withan essentially one-dimensional problem [8–25]. Previousstudies have focussed on various mechanisms that controlmean length of such protrusions. But, to our knowledge,their temporal fluctuations have received very little at-tention [8, 18].The temporal fluctuation of the lengths of long cellprotrusions may have important implications in their bi-ological functions [26–31]. For example, if the fluctuatinglength of a sensory cilium falls below a certain thresholdit may not be able to pick up molecular signals floatingbeyond that threshold. When such a situation arises, theshortened cilium becomes silent; the consequent ‘ fading ’of chemical signals received by the cell would be remi-niscent of the phenomenon of fading of electromagneticsignals that has been studied extensively [32, 33] sincethe pioneering work of Rice [34, 35]. The silent cilium ∗ E-mail: [email protected] regains its sensory capability only after its length growsabove that threshold because of fluctuations. Similarly,growing beyond an upper threshold may lead to excessiveexposure to the surrounding environment. This will addto the metabolic cost and disturb the energy budget ofthe cell [36].The cell protrusions that motivate this work grow andshrink by assembling or disassembling materials at theirdistal tips [18]. But, these do not have the machineries ofprotein synthesis inside the protrusion. Consequently, allthe building materials required for elongation of the pro-trusion need to be imported from the cell body and trans-ported to the distal tip (anterograde transport). Simi-larly the building materials released from the shrinkingdistal tip also have to transported back to the cell body(retrograde transport). Transport in both these direc-tions are driven by molecular motors that are poweredby a chemical fuel called ATP. Because of the intrinsicstochasticities of the ongoing turnover of the buildingblocks and incorporation of new materials [8, 37–40], thesteady state length keeps fluctuating about the mean (av-erage length) even after the protrusion attains its steadystate [41–44]. However, the theory developed in this cur-rent paper is not limited to such potrusions. It is ap-plicable to all such protrusion and filaments which growor shrink by adding or removing monomers from theiredges (see Fig.1(a)). For a list of such one dimensionalstructures, readers are referred to the review article byMohapatra et al [8].Steady state length is achieved by striking a balancebetween the assembly and disassembly of the protrusion[8, 18, 38]. If both these rates are constants and inde-pendent of the instantaneous length of the protrusionthen, barring an exceptional condition, balance of as-sembly and disassembly cannot be achieved. To attain a r X i v : . [ q - b i o . S C ] A ug Assembly r+ ( ℓ )Disassembly r- ( ℓ +1) (a)(b) (b1) (b2) (b3) 〈 ℓ ss 〉 〈 ℓ ss 〉 〈 ℓ ss 〉 Length control by assembly & Length control by assembly Length control by disassembly R a t e Length Length Length R a t e R a t e a ss e m b l y a ss e m b l y assembly d i s a ss e m b l y d i s a ss e m b l y d i s a ss e m b l y disassembly FIG. 1:
Length control in generic protrusions andfilaments : (a) Generic protrusions and filaments are mod-elled as lattice chains which can grow or shrink by adding orremoving monomers from its end. (b) A controlled length ofthe structures emerges when the assembly rate is balanced bythe disassembly rate. Therefore, for maintaining finite lengthof the structure (b1) either both the assembly and disassem-bly rates (b2) or the assembly rate (b3) or the disassemblyrate should be dependent on the length of the structure. a stationary (i.e., time-independent) mean length, eitherthe assembly rate should decrease with increasing lengthor the disassembly rate should increase with increasinglength or both these phenomena must occur. Accord-ingly, the protrusions can be divided into three classes:(a) assembly controlled, (b) disassembly controlled and(c) both assembly and disassembly controlled [8] (see Fig.1(b)).Length-dependence of the rates of growth and shrink-age of a protrusion can arise from length-dependentregulation of the corresponding anterograde and retro-grade transport processes [11]. Alternatively, length-dependence of the shrinking rate of a protrusion can arisefrom a length-dependent depolymerization of the stiff fil-aments, that form the cylindrical scaffolding, by special-ized enzymes called depolymerases [16, 24, 45]. We willsolve the problems for such generic protrusions. However,to get an intuitive feel for the actual numbers, we willapply the analytical formulae derived here exclusively toeukaryotic flagellum which is an example of assembly-controlled protrusion.The fluctuations of the protrusion length can be viewedas one-dimensional random excursions of its distal tip. Ifa cell protrusion has certain upper and lower thresholdfor its lengths, the random excursions of the tip can beformulated as level crossing problems [32, 47–63]. Thespecific questions we address are : (a) How frequentlydoes the length cross the upper and lower thresholds byfluctuations ? (b) What is the duration for which a pro-tusion remains beyond these thresholds ? (c) What isthe maximum or the minimum length the protrusion cangrow or shorten to during its lifetime ? Having an esti-mate for such rates, timescales and lengthscales can helpus to get deeper insights into the nature of fluctuations of the length of cell protrusions.The manuscript is organised in the following manner.First, we will discuss how to model the assembly andlength control of a growing protrusion or filament. Itwill be followed by the formulation of equations for cap-turing the stochastic growing shrinking dynamics of thesedynamic structures. Thereafter, we show how the fluc-tuating length of the protrusions and filaments in thesteady state can be described by an Ornstein-Uhlenbeck(OU) process. The introduction of length control in eu-karyotic flagellum will be followed by the derivation ofuseful expressions for mean passage times, random excur-sions and extremal excursions. Finally, using the derivedexpressions we estimate the level crossing quantities foreukaryotic flagellum. The details of the calculations havebeen provided in the appendices.
II. EQUATIONS FOR PROTRUSION KINETICSAND FORMAL STEADY STATE SOLUTIONS
The protrusions are modelled here as one-dimensionallattices (i.e., linear chains) of equispaced discrete sites[8, 18]. The growing and shortening of the structures iscaptured by the addition and removal of monomers fromthe distal end of the lattice as shown in Fig.1(a).For these structures to achieve a controlled length, therate of growing by incorporating new monomers shouldbe balanced by the rate of shortening by discarding themonomers from the structure. This scenario is possi-ble only if both these rates or at least one of them islength dependent. The controlled length emerges wherethe curves intersect and balance each other (see Fig.1(b)).The cell assembles protrusions and the filaments fromscratch and the average length keep increasing during thegrowing phase. At some point of time, the average lengthceases to change further on the attainment of the steadystate. However, the instantaneous length keeps fluctuat-ing about the average length and these length fluctua-tions are known as steady state length fluctuations.For the quantitative description of the growing andshrinking dynamics of the generic protrusions, we treatit as a stochastic process L ( t ) where the stochastic ki-netics of the length of the protrusion are assumed to beMarkovian. Let P L ( (cid:96), t ) be the probability that the pro-trusion is of length L ( t ) = (cid:96) at time t . For simplicitywe consider that each monomer is of unit length, so aprotrusion of length (cid:96) consists of (cid:96) number of monomers. A. Master Equation for stochastic kinetics:discrete length
The master equation governing the stochastic kinetics[64–66] of the length evolution of the protrusion is givenby dP L ( (cid:96), t ) dt = r − ( (cid:96) + 1) P L ( (cid:96) + 1 , t ) − r + ( (cid:96) ) P L ( (cid:96), t ) for (cid:96) = 0(1a) dP L ( (cid:96), t ) dt = r + ( (cid:96) − P L ( (cid:96) − , t ) + r − ( (cid:96) + 1) P L ( (cid:96) + 1 , t ) − ( r + ( (cid:96) ) + r − ( (cid:96) )) P L ( (cid:96), t ) for (cid:96) = 2 to N − dP L ( (cid:96), t ) dt = r + ( (cid:96) − P L ( (cid:96) − , t ) − r − ( (cid:96) ) P L ( (cid:96), t ) for (cid:96) = N (1c) where the rate of growing ( r + ( (cid:96) )) and shortening ( r − ( (cid:96) ))are length dependent. N denotes the maximum numberof monomers which can be incorporated into the protru-sion. The motivation for introducing an upper cutoff N for the allowed length of the protrusion is that the poolof structural materials are supplied by the cell body andthis supply is finite [5, 6, 68]. Besides, N being a pa-rameter, its magnitude can be varied and adjusted whenapplied to a specific protrusion.The steady state solution of the master equation (1) isgiven by P s ( (cid:96) ) = P (0) (cid:96) (cid:89) j =1 r + ( j − r − ( j ) (2)where P (0) = (cid:18) N (cid:88) (cid:96) =1 (cid:96) (cid:89) j =1 r + ( j − r − ( j ) (cid:19) . (3)This gives the distribution of protrusion length in thesteady state. B. Fokker-Planck equation for stochastic kinetics:continuous length
Next we take the continuum limit [64–66] in which thelength of the protrusion is represented by a continuousvariable y which is defined by y = (cid:96)/N. (4)So the range of allowed values of y is 0 ≤ y ≤
1. Be-sides, since both (cid:96) and N are dimensionless so is y . Inthis continuum limit, the probability P L ( (cid:96), t ) reduces to p ( y, t ) which denotes the probability that the protrusionlength is y at time t . Carrying out the standard Kramers-Moyal expansion of the master equation (1b) governingthe length of the protrusion, we obtain the correspondingFokker-Planck equation ∂p ( y, t ) ∂t = − ∂∂y (cid:20) R − ( y ) p ( y, t ) (cid:21) + 12 N ∂ ∂y (cid:20) R + ( y ) p ( y, t ) (cid:21) (5) where R − ( y ) = r + ( y ) − r − ( y ) & R + ( y ) = r + ( y ) + r − ( y ) . (6) Let us rewrite the Fokker-Planck equation (equation(5)) in the following form ∂p ( y, t ) ∂t = − ∂∂y (cid:18) R − ( y ( t )) p ( y, t ) − N ∂R + ( y ( t )) p ( y, t ) ∂y (cid:124) (cid:123)(cid:122) (cid:125) J (cid:19) (7) where the underlined term is the flux J . In the steadystate ∂p ( x,t ) ∂t = 0 and this indicates that J is constant.Setting J = 0 , we get the steady state solution p s ( y ) ofthe Fokker Planck equation as p s ( y ) = N R + ( y ) exp (cid:18) N (cid:90) R − ( y (cid:48) ) R + ( y (cid:48) ) dy (cid:48) (cid:19) (8)where N is the normalization constant. C. Rate equation for deterministic kinetics:continuous length
From the master equations governing the stochastictime evolution of the length of the protrusions, we getthe corresponding rate equation [64–66] dL ( t ) dt = r + ( L ( t )) − r − ( L ( t )) . (9)where L ( t ) = (cid:104) (cid:96) ( t ) (cid:105) = (cid:80) N(cid:96) =0 P L ( (cid:96), t ). The steady statesolution of the rate equation (9) can be obtained by solv-ing r + ( L ss ) = r − ( L ss ) (10)which gives the measure of protrusion length L ss in thesteady state. III. MAPPING PROTRUSION LENGTHFLUCTUATION ONTO OU PROCESS
For the Fokker Planck equation (5), the correspondingstochastic differential equation (SDE) or the Langevinequation [64–66] describing the evolution of the length ofthe protrusion is given by (see appendix A for the details) dy = R − ( y ) dt + 1 √ N (cid:112) R + ( y ) dW ( t ) (11)where W ( t ) is the Gaussian white noise (a Wiener pro-cess) with dW ( t ) distributed according to a Gaussianprocess with mean and covariance given by (cid:104) dW ( t ) (cid:105) = 0 (mean) (cid:104) dW ( t ) dW ( s ) (cid:105) = δ ( t − s ) dt ds (covariance) (12)In the limit N → ∞ , from (11) we recover the determin-istic equation for y which, after substitutions from (6)and (10), yields the fixed point y ∗ .The SDE (11) describing the stochastic evolution ofthe protrusion length involves Gaussian like fluctuationsof order 1 / √ N about the deterministic trajectory. Wemake a change of variable from y to x by defining y − y ∗ = x/ √ N where x is a measure of the deviation of y from its steady-state value y ∗ . Thus, fluctuations of y around y = y ∗ is equivalent to that of x around x ∗ = 0.Accordingly the functions of R ± ( y ) of y get transformedto the functions R ± ( x ) of x . Formally Taylor expandingthe RHS of (11) to the lowest order in 1 / √ N (the so-called linear noise approximation [64]) yields dx = R − ( x ) dt + (cid:112) R + ( x ) dW ( t ) (13)where R − ( x ) = R (cid:48)− ( y ∗ ) x and R + ( x ) = R + ( y ∗ ) (14)The details of deriving equation (13) from equation (11)are given in appendix B.Note that the transformed Langevin equation (equa-tion (13)) has the formal structure of a standard Orn-stein - Uhlenbeck (OU) process [69] for which the SDEreads as dx = − κx dt + √ D dW ( t ); (15)and noise satisfies (12). As is well known, the Fokker-Planck equation corresponding to the SDE (15) is givenby ∂p ( x, t ) ∂t = ∂∂x (cid:20) κx p ( x, t ) (cid:21) + D ∂ p ( x, t ) ∂x (16)The Fokker-Planck equation corresponding to thetransformed Langevin equation (13) for protrusion lengthis ∂p ( x, t ) ∂t = ∂∂x (cid:20) R − ( x ) p ( x, t ) (cid:21) + 12 ∂ R + ( x ) p ( x, t ) ∂x = ∂∂x (cid:20) − R (cid:48)− ( y ∗ ) x p ( x, t ) (cid:21) + R + ( y ∗ )2 ∂ p ( x, t ) ∂x (17)Identifying κ = − R (cid:48)− ( y ∗ ) & D = R + ( y ∗ ) (18)we can map the Fokker-Planck equation (17) into OUprocess (equation (16 )).Rescaling the protrusion length ( y − y ∗ = x/ √ N ) andmapping the phenomena into OU process (equation (16)and (17)) allows us to view the movement of the tipabout the mean length due to the growing and short-ening of the protrusion length as a moving Brownianparticle subjected to linear drift (see Fig.2). Note thatthe allowed range of values of x is −√ N ≤ x ≤ √ N .Now to understand the steady state length fluctuationsof the protrusion or the dynamics of x , we can usethe techniques developed under the framework of OUprocess to study the dynamics of the Brownian particlewhich is subjected to linear drift. In table I, we haveconsidered three class of protrusions whose growing and Assembly rate Disassembly rate y*y L y U T i m e Protrusion length y*y L y U FIG. 2:
Fluctuating protrusion length :
Protrusionlength evolves in time by adding and removing precursorsfrom its tip. The fluctuating protrusion length can be treatedas a stochastic process. Due to length dependent assemblyand disassembly rate, the protrusion length keeps fluctuatingabout its mean value in steady state. shrinking rates are either constant or change linearlywith protrusion length. We have summarized theexpressions of rates for master equation, rate equations,Fokker-Planck equation and the expression of κ and D when the length fluctuations for these structures aremapped into OU process.Solutions : The probability distribution p ( x, t | x , t )by solving the Fokker Planck equation (16) [64–66, 69]reads as p ( x, t | x , t ) = (cid:114) κπD [1 − e − κt ] exp (cid:18) − κ ( x − x e − κt ) D [1 − e − κt ] (cid:19) (19) where x is the initial position of the protrusion tip.If the initial mean position is (cid:104) x (cid:105) , the mean position (cid:104) X ( t ) (cid:105) evolves as (cid:104) X ( t ) (cid:105) = (cid:104) x (cid:105) e − κt (20)and the variance reads as (cid:104) ( X ( t ) − (cid:104) X ( t ) (cid:105) ) (cid:105) = D κ (1 − e − κt ) (21)The steady state distribution is given by p ss ( x ) = (cid:114) κπD exp (cid:18) − κx D (cid:19) (22)and the variance in steady state is given by σ = D κ . (23) Quantities Assembly Controlled Disassembly Controlled Assembly-Disassembly ControlledMaster equation r + (ℓ) 𝜆 A (N-ℓ) 𝜆 A N 𝜆 A (N-ℓ)r - (ℓ) 𝜇 A N 𝜇 D ℓ 𝜇 AD ℓ Rate equation 𝜆 A (1 - 〈 ℓ ( t ) 〉 ) 𝜆 D 𝜆 AD (1 - 〈 ℓ ( t ) 〉 ) 𝜇 A 𝜇 D 〈 ℓ ( t ) 〉 𝜇 AD 〈 ℓ ( t ) 〉 Fokker -Planck Equation R + ( y ) 𝜆 A (1 - y )- 𝜇 A 𝜆 D - 𝜇 D y 𝜆 AD (1 - y )- 𝜇 AD y R_ ( y ) 𝜆 A (1 - y )+ 𝜇 A 𝜆 D + 𝜇 D y 𝜆 AD (1 - y )+ 𝜇 AD y Mapping into Ornstein-Uhlenbeck process 𝜅 𝜆 A 𝜇 D 𝜆 AD + 𝜇 AD D 𝜇 A 𝜆 D [( 𝜆 AD 𝜇 AD )/( 𝜆 AD + 𝜇 AD )] TABLE I: Summarizing the terms of master equation, rateequation and Fokker-Planck equation for assembly controlled,disassembly controlled and assembly-disassembly controlledprotrusions whose rates of assembly and disassembly ratesare either constant or linearly changing with the length ofthe protrusion and listing the expressions of κ and D whichare required for mapping the evolution of protrusion lengthinto Ornstein-Uhlenbeck process. Note that λ s and µ s arethe phenomenological constants which can be extracted fromexperimental data. Looking at the expression for p ( x, t | x , t ) and (cid:104) X ( t ) (cid:105) ,the inverse κ − is actually the timescale with which theprotrusion length relaxes to its steady state [69]. Thelinear drift velocity κx is restoring in nature and al-ways drives the protusion tip towards the mean position x ∗ , which is the origin ( x ∗ = 0) for the process X ( t ).More the deviation from the mean position ( x − x ∗ = x ),stronger is the drift κx .The protrusion is growing by adding monomers toits tip and is shortening by removing monomers fromits tip (Fig.2). When the assembly rate balances thedisassembly rate, a controlled length of the protrusionemerges. If the protrusion grows beyond its controlled(or mean) length, the disassembly rate becomes domi-nant and forces the protrusion to regain its correct / ideallength by shortening (Fig.2). Similarly, if the protrusionshortens further, the dominant assembly rate helps inreclaiming the correct length by growing. Mapping thegrowing - shortening dynamics of the protrusion lengthinto the OU process further clarifies the physical picture.It says that the tip keeps doing a random walk by addingand removing subunits but there is always a drift actingon it which drives the protrusion length towards its meanlength (or the fixed point x ∗ ).Whenever we talk about a controlled system undergo-ing fluctuation, there must be certain upper ( x U ) and lower ( x L ) thresholds crossing which may have certainimplications (Fig.2). In the coming sections, we will dis-cuss the passage times for hitting the the thresholds, thetime spent beyond and below these thresholds and theextremum length to which the protrusion can grow orshrink to in a certain interval of time. IV. A CASE STUDY WITH AN ASSEMBLYCONTROLLED PROTRUSION
For case study, let us consider a protrusion whoselength is controlled by assembly. The rate of assem-bly ( r + ( (cid:96) )) falls exponentially with the increasing lengthwhereas the rate of disassemly ( r − ( (cid:96) )) remains constantthroughout. Such choice of rates for our model protru-sion is motivated by the eukaryotic flagellum [18, 38].The length dependent growing rate ( r + ( (cid:96) )) and thelength independent shortening rate ( r − ( (cid:96) )) for our modelprotrusion are given by r + ( (cid:96) ) = Ae − C(cid:96) and r − ( (cid:96) ) = B (24)where A , B and C are phenomenological constants. If (cid:96) is converted into proper length by multiplying it with thelength ∆ (cid:96) of the subunits of the protrusion, then C mustalso be divided by ∆ (cid:96) to keep the exponential dimen-sionless. More about the internal structure, mechanismof acquiring length feedback and how different cytoskele-tal and interflagellar components coordinate for assem-bling flagellum of controlled length will be discussed insection VIII. A more detailed description is provided inour recently published paper [18]. A. Steady state: rate equation approach
Using the general form (10) of the steady-state solu-tion, for our special case (24) under study, we get thesteady state mean length to be L ss = 1 C (cid:96)og (cid:18) AB (cid:19) (25)by solving the equation Ae − CL ss = B . B. Steady state: master equation approach
The steady state solution of the master equation (equa-tion (1)) governing the length of the assembly controlledprotrusion is obtained by substituting the rates men-tioned in equation (24) in the expression mentioned inequation (2) and (3). The following solution of the mas-ter equation P s ( (cid:96) ) = P (0) (cid:18) AB (cid:19) (cid:96) e − (cid:96) ( (cid:96) +1) C/ (26)where P (0) = (cid:18) N (cid:88) (cid:96) =1 (cid:18) AB (cid:19) (cid:96) e − (cid:96) ( (cid:96) +1) C/ (cid:19) − (27)gives the distribution of the length of the protrusion inthe steady state. In terms of A , B and C , the steady statesolution of the rate equation (9) governing the length (cid:104) (cid:96) ss (cid:105) = 1 C (cid:96)og (cid:18) AB (cid:19) . (28)gives the measure of average length of the assembly con-trolled protrusion in the steady state. Note that the meanlength (28) of the protrusion is identical to the expres-sion (25) of the steady-state length obtained from therate equation thereby confirming mutual consistency ofthe two approaches. C. Steady state and Fluctuations aroundsteady-state: Fokker-Planck approach
For the Fokker-Planck equation (5) governing the evo-lution of the length of the assembly controlled protrusion, R + ( y ) and R − ( y ) are given by R + ( y ) = Ae − Cy + B and R − ( y ) = Ae − Cy − B. (29)Substituting the expressions (29) for R ± ( y ) into the gen-eral form (8) for the steady state solution, we get p s ( y ) = N e − N Ψ( y ) Ae − Cy + B (30)where the pseudopotential Ψ( y ) is given byΨ( y ) = 2 (cid:90) R − ( y (cid:48) ) R + ( y (cid:48) ) dy (cid:48) = 2 (cid:18) (cid:96)og (cid:20) Ae − Cy + Be − Cy ( A + B ) (cid:21) − y (cid:19) (31)The most probable value of y that, by definition, corre-sponds to an extremum of the pseudopotential, lies at y ∗ .Exploiting the definition y = (cid:96)/N , we get the most prob-able length (cid:96) mp = N y ∗ of the protrusion in the steadystate to be (cid:96) mp = 1 C (cid:96)og (cid:18) AB (cid:19) , (32)which is identical to the expressions (25) and (28) of themean length of the protrusion in the steady-state. Al-ternatively, from equation (11) in the limit N → ∞ weget the steady-state length to be given by R − ( y ss ) = 0 = Ae − Cy ss − B , which yields the same expression for thesteady-state length as obtained from all the other ap-proaches discussed above. The steady state solution ofthe master equation, the rate equation and the FokkerPlanck equation for the assembly controlled protrusionhas been plotted in Fig.3. ℓ y P s ( ℓ ) p s ( y ) 〈 ℓ ss 〉 FIG. 3:
Steady state solution for model protrusion :The steady state solution of the master equation ( P s ( (cid:96) ) as afunction of (cid:96) ) is represented by dots and that of Fokker Planckequation ( p s ( y ) as a function of y ) is represented by the solidblue line. The steady-state length obtained from the rateequation ( L ss ), which is identical to the mean length ( (cid:104) (cid:96) ss (cid:105) )obtained from the master equation and the most probablelength ( (cid:96) mp ) obtained from the Fokker-Planck equation, ismarked by the dotted vertical line. Values of A = 1 . × − , B = 4 . × − and C = 2 . × − are used for makingthe plots. D. Mapping onto OU Process and Fluctuationsaround steady-state
Substituting the expressions (29) for R ± ( y ) in the spe-cific case of assembly-controlled protrusions into the gen-eral definitions (18) of κ and D we get κ = − R (cid:48)− ( y ∗ ) = BC and D = R + ( y ∗ ) = 2 B. (33)The Fokker-Planck equation for the transformedLangevin equation (equation (13)) describing the dynam-ics of the flagellar tip fluctuating about its mean positionis given by ∂p ( x, t ) ∂t = ∂∂x (cid:20) BC (cid:124)(cid:123)(cid:122)(cid:125) κ x p ( x, t ) (cid:21) + B (cid:124)(cid:123)(cid:122)(cid:125) D/ ∂ p ( x, t ) ∂x (34)Rescaling the length ( y − y ∗ = x/ √ N ) of the assem-bly controlled protrusion and mapping it into OU processallows us to analyze the system using the techniques de-veloped for OU process.The length fluctuations, for the initial condition x ,are characterized by the probability distribution p ( x, t | x , t )= (cid:115) C π [1 − e − BCt ] exp (cid:18) − C ( x − x e − BCt ) − e − BCt ] (cid:19) (35)which, in the limit t → ∞ , reduces to the steady-statedistrubution p ss ( x ) = (cid:114) C π exp (cid:18) − Cx (cid:19) . (36) V. STATISTICS OF PASSAGE TIMES
One of the central problems of level crossing is esti-mating the time the system takes to reach a particularboundary for the first time. In the context of a fluctu-ating protrusion, our interest lies in estimating the timethe tip of the protrusion takes to cross a critical valueof its length (or, equivalently, the hypothetical Brownianparticle takes to cross a threshold boundary) for the firsttime [46, 55, 57, 60, 62].Suppose, as shown in figure 4(a), initially the positionof Brownian particle (tip) ( x ) lies between two thresh-olds x U and x L ( x L ≤ x ≤ x U ). One of our aims isto calculate the time to escape (or exit) the safe zone,which is bounded by the upper ( x U ) and the lower ( x L )thresholds, by crossing either of the two thresholds. Thetime the Brownian particle takes to hit either of the twothreshold for the first time defines its first exit time cor-responding to the given initial condition. In case thereis a single threshold x th ( x th > x ∗ ) of interest, there aretwo ways of hitting it (see figure 4(b1-b2)). If the initialposition x lies below the threshold x th as shown in fig-ure 4(b1), then it must move upward to hit the thresholdlength x th and the first time it hits the threshold frombelow is known as the first upcrossing time. On the oth-erhand, if x lies above x th (figure 4(b2)), it must movedownward to hit the threshold x th ; the first time it hitsthe threshold x th from above is the first downcrossingtime. As we have mapped the movement of the tip ontothat of a hypothetical particle that executes an OU pro-cess, now we will calculate the mean exit, upcrossing anddowncrossing times for the system using the techniquesfor OU process which are nicely reviewed by Masoliver[46].As is well known, calculation of mean first passagetimes are normally more convinient if one uses back-ward Fokker-Planck equation, rather than the forwardFokker-Planck equation given by equation (16) [69]. Forthe generic model under our consideration, the backwardFokker-Planck equation is given by − ∂p ( x, t | x , t ) ∂t = − κx ∂p ( x, t | x , t ) ∂x + D ∂ p ( x, t | x , t ) ∂x (37) As both the drift κx and the diffusion D are not de-pendent on time t and t explicitly, p ( x, t | x , t ) dependson time only through the difference t − t . Therefore,the evolving protrusion length is considered to be tempo-rally homogenous system for which the backward Fokker-Planck equation is given by ∂p ( x, t | x , t ) ∂t = − κx ∂p ( x, t | x , t ) ∂x + D ∂ p ( x, t | x , t ) ∂x (38) as by chain rule ∂ t = − ∂ t .The probability that the particle located at x at time t escapes the safe zone, bounded by the upper ( x U ) and the lower ( x L ) thresholds, for the first time at time t isgiven by E ( t ; x U , x L | x , t ) = 1 − (cid:90) x L x U p ( x, t | x , t ) dx (cid:124) (cid:123)(cid:122) (cid:125) probability that the particleis lying between x U and x L (39)The probability of hitting a threshold x th is closelyrelated to the escape probability defined in equation (39).Let H ( t ; x th | x , t ) denote the probability of hitting thethreshold x th at time t , given that the particle was at x at time t . In case x th > x , the hitting (or upcrossing)probability is given by H U ( t ; x th | x , t ) = E ( t ; x U = x th , x L = −∞| x , t ) (40)which is like escaping a semi-infinite interval ( −∞ , x th ).On the other hand, if x th < x ,the hitting (or downcross-ing) probability is given by H D ( t ; x th | x , t ) = E ( t ; x U = ∞ , x L = x th | x , t ) (41)which is like escaping a semi-infinite interval ( x th , ∞ ). Inthe following subsections, we will present the expressionsfor the mean exit time from the region bounded by twothresholds, the mean hitting time for a given threshold x th and discuss their implications. A. Escaping the safe zone
The mean escape time taken by the protrusion to growor shrink beyond x U and x L respectively and escaping thesafe zone is denoted by T E ( x U , x L | x , t ) = (cid:90) t p E ( t, x U , x L | x , t ) dt (42)where p E ( t, x U , x L | x , t ) is the pdf corresponding to theescape probability E ( t ; x U , x L | x , t ). In appendix C 1,we have shown that T E ( x U , x L | x , t ) satifies the follow-ing ordinary differential equation − κx d T E ( x U , x L | x , t ) dx + D d T E ( x U , x L | x , t ) dx = − T E ( x U , x L | x U , t ) = 0 = T E ( x U , x L | x L , t ) . (44)On solving it, we get the expression for the mean escapetime T E ( x U , x L | x , t ) which reads as x U x L x x th x*xx th x*x Exit time Upcrossing time Downcrossing time
Escaping the safe zone Hitting the threshold (a) (b)(b1) (b2) x Rescaled mean exit time (c)
Mean upcrossing timeMean downcrossing time R e sc a l ed m ean h i tt i ng t i m e x (d) κ 𝓣 𝓗 ( x ) κ 𝓣 𝓔 ( x ) - -
20 0 20 400.20.40.60.81.01.21.4
FIG. 4:
Statistics of passage time : (a) Space-time diagram for the hypothetical Brownian particle which captures themovement of the tip of the protrusion. It escapes the region bounded by an upper x U and lower x L threshold by crossing eitherof the thresholds and the time taken is the known as the exit time. (b) Space-time diagram for the Brownian particle which liesbelow (b1) and above (b2) a threshold x th and time taken to hit the threshold is given by upcrossing (b1) and downcrossing (b2)time respectively. (c) Mean exit time for seven different pairs of ( x U , x L ). (d) Mean upcrossing (solid line) and downcrossing(dashed lines) time for hitting two different threshold x th = 10 (green) and x th = 15 (red). Value of κ = 9 . × − and D = 8 . × − . T E ( x U , x L | x , t ) = (cid:20) D (cid:18) erfi (cid:18) √ κx L √ D (cid:19) − erfi (cid:18) √ κx U √ D (cid:19)(cid:19) (cid:21) − × (cid:20) x (cid:18) erfi (cid:18) √ κx U √ D (cid:19) − erfi (cid:18) √ κx L √ D (cid:19)(cid:19) F (cid:18) ,
1; 32 , κx D (cid:19) + x L (cid:18) erfi (cid:18) √ κx √ D (cid:19) − erfi (cid:18) √ κx U √ D (cid:19)(cid:19) F (cid:18) ,
1; 32 , κx L D (cid:19) + x U (cid:18) erfi (cid:18) √ κx L √ D (cid:19) − erfi (cid:18) √ κx √ D (cid:19)(cid:19) F (cid:18) ,
1; 32 , κx U D (cid:19) (cid:21) (45) where the function erfi( z ) is the imaginary error function[70] whose series about z = 0 is given byerfi( z ) = 1 √ π (cid:18) z + 23 z + 15 z + 121 z + ... (cid:19) (46)and F ( a , a ; b , b ; z ) is the generalized hypergeomet-ric function [70] whose series about z = 0 is given by F ( a , a ; b , b ; z ) = ∞ (cid:88) j =0 ( a ) j ( a ) j ( b ) j ( b ) j z j j ! (47)In Fig.4(c), we have plotted the mean exit time T E ( x U , x L | x , t ) as a function of initial position x forseven different pairs of ( x U , x L ). B. Hitting the thresholds
If there is a single threshold ( x th ) of interest, the ap-propriate quantity is the mean hitting time T ( x th | x , t )which is the mean time taken to hit the threshold x th for the first time if initially its length is x . The meanhitting time is given by T H ( x th | x , t ) = (cid:90) ∞ t p H ( t ; x th | x , t ) dt (48)where p H ( t ; x th | x , t ) is the pdf associated with the hit-ting probability H ( t ; x th | x , t ). The steps for derivingthe expression of T H ( x th | x , t ) are given in the appendixC 2.For x th > x (Fig.4(b1)), the mean hitting or upcrossing time for hitting the threshold is T HU ( x th | x , t ) = 1 κ (cid:90) κx th /Dκx /D F (1 , / , z ) dz = √ πκ (cid:90) √ κ | x th | / √ D √ κ | x | / √ D e y erf( y ) dy = 1 D (cid:20) x F (cid:18) ,
1; 32 , κx D (cid:19) − x
20 2 F (cid:18) ,
1; 32 , κx D (cid:19)(cid:21) (49)For x th < x (Fig.4(b2)), the mean hitting or downcrossing time for hitting the threshold is T HD ( x th | x , t ) = 12 κ (cid:90) κx /Dκx th /D U (1 , / , z ) dz = √ πκ (cid:90) √ κ | x | / √ D √ κ | x th | / √ D e y erfc( y ) dy = 12 κ (cid:20) π erfi (cid:18) √ κx √ D (cid:19) − π erfi (cid:18) √ κx th √ D (cid:19)(cid:21) − D (cid:20) x
20 2 F (cid:18) ,
1; 32 , κx D (cid:19) − x th F (cid:18) ,
1; 32 , κx th D (cid:19)(cid:21) (50)In figure 4(d), we have plotted the mean upcrossing (solid line) and downcrossing time (dashed line) for hitting twodifferent thresholds x th = 10 (green line) and x th = 15 (red line). In equation (49) and equation (50), F (1 , / , z )and U (1 , / , z ) are the generalised Kummer’s functions [70]. C. Implications
1. Mean Escape time
The mean escape time plotted as a function of x in Fig.4(c) varies in a non-monotonous manner with the initialposition x . The curves in Fig.4(a) exhibit a maximum at x max which is given by x max = (cid:114) Dκ × erf − (cid:20) κπD (cid:16) x L F (cid:16) , , x L κD (cid:17) − x U F (cid:16) , , x U κD (cid:17)(cid:17)(cid:16) erfi (cid:16) √ κx L √ D (cid:17) − erfi (cid:16) √ κx U √ D (cid:17)(cid:17) (cid:21) (51)In order to get physical insight into the implications of the exact mathematical expression (48) of T E ( x U , x L | x , t ),let us consider a few special cases all of which place the thresholds symmetrically at equal distances from the originlocated at x = 0. The origin x = 0 coincides with the mean position of the Brownian particle ( x ∗ ) in the steady-statebecause x ∝ ( y − y ∗ ). Special case I : Suppose, the two thresholds are placed symmetrically at x U = a and x L = − a , respectively. For thisparticular positions of the pair of thresholds, the expression (48) for T E ( x U , x L | x , t ) simplifies to T E ( a, − a | x ,
0) = 1 D (cid:20) a F (cid:18) ,
1; 32 , a κD (cid:19) − x
20 2 F (cid:18) ,
1; 32 , κx D (cid:19) (cid:21) (52)where x -dependence arises only from x appearing inthe second term within the square bracket and renders T E ( a, − a | x ,
0) symmetric with respect to change of signof x . This symmetry arises from the fact that the dis-tances of the upper and lower thresholds from x = b are identical to those to the lower and upper thresholds,respectively, from x = − b . Moreover, not surprisingly,in this symmetric case, the general expression (51) for x max coincides with the origin x = 0. In the more gen-eral asymmetric case, T E ( x U , x L | x ,
0) is not symmetric.Specializing further to x = 0, only the a -dependent first term within the square bracket in (52) survives while thesecond term vanishes. Special case II : A particular lengthscale σ associatedwith the problem is σ = (cid:114) D κ . (53)which is obtained by taking the square root of the vari-ance (mentioned in equation (23)) associated with thesteady state distribution of X ( t ) given by equation (22).0Using the series F (cid:18) ,
1; 32 , z (cid:19) = 1 + z z
45 + O ( z ) (54)we get the approximate expression for T E ( a, − a | x , t )when both x and a are much smaller than σ to be T E ( a << σ, − a << − σ | x , t ) ≈ D ( a − x ) (55)which explains the inverse parabolic form appearing inFig.4(c). In this special case the mean exit time dependsonly on D and is independent of κ . The magnitude of thedrift in the region − σ < x < σ is negligible. Therefore, itis diffusion only which drives the escape from this region. Special case III : Another special case is when x U = σ, x L = − σ . If in this case x = 0, then T E ( σ, − σ | x , t ) = 0 . κ (56)i.e., the mean exit time is purely a function of the relax-ation time κ − , irrespective of the magnitude of D . Special case IV : Suppose the initial position x is veryclose to one of the boundaries, say x = a while the otherboundary is located at x = − a . By the Taylor seriesexpansion of the expression T E ( a, − a | x ,
0) about x = a ,we getlim x → a T E ( a, − a | x , t ) = (cid:114) πDκ e κa /D erf (cid:18) a (cid:114) κD (cid:19) ( a − x )(57)i.e., the mean exit time is a linear fuction of the distance( a − x ) between the initial position and the boundary at x = a . In equation (57) erf( z ) denotes the error function[70] whose series about z = 0 is given by erf ( z ) = 2 √ π ∞ (cid:88) j =0 ( − j z j +1 j !(2 j + 1) (58)
2. Mean Hitting time
The results of mean hitting time (Fig.4(d)) are inter-esting. Naively, one might expect the mean time for up-crossing the threshold at x th from x > x th to be identicalto mean time for downcrossing the same threshold from x < x th if the distance | x − x th | is same in both thecases. But, that is not true, as the detailed analysis ofour results (49) and (50) establish. We plot the results numerically for two different val-ues of the threhold, namely x th = 10 and x th = 15 inFig.4(d). For lower threshold x th = 10, the mean down-crossing time is longer than the mean upcrossing time(if | x − x th | is same in both the cases) as indicated bythe horizontal bars marked on the curve for the mean hit-ting time for the threshold x th = 10 (green curve in figure4(d)). The opposite is seen for higher threshold x th = 15.In this case, the mean upcrossing time is longer than themean downcrossing time as indicated by the horizontalbars marked on the curve for the mean hitting time forthe threshold x th = 15 (red curve in figure 4(d)). Thereason behind this is the restoring linear drift acting onthe tip. In case of lower threshold, the linear drift drivingthe tip towards mean is weaker. Therefore, it takes longertime for downcrossing the threshold rather than upcross-ing it. On the other hand, for higher thresholds, therestoring drift is much stronger. This makes upcrossinga difficult task compared to the downcrossing the thresh-old. Therefore, the mean upcrossing time is much longerthan the mean downcrossing time for a higher threshold.The expression in equation (49) for mean hitting timeof the threshold by upcrossing it is same as the meanescape time given by expression in equation (52) exceptthat a in the later expression (equation (52)) is replacedby x th in the former one (equation (49)). So, in all specialcases and special limits, the expression for mean hittingtime T HU ( x th | x , t ) resembles T E ( x ; a, − a ) which aresummarised in equation (55-57). On the other hand, theexpression for mean hitting time for hitting a thresholdby downcrossing it is different so will be the expressionfor different special cases and in special limits.In order to gain insight into the physical implicationsof the mathematical expressions (49) and (50), as before,we consider simple situations. If initially x = 0, themean upcrossing time to any arbitrary threshold x th is T HU ( x th | ,
0) = x th D F (cid:18) ,
1; 32 , κx th D (cid:19) (59)whereas if the threshold is x th = 0, the mean downcross-ing time from any initial location x is T HD (0 | x ,
0) = π κ erfi (cid:18) √ κx √ D (cid:19) − x D F (cid:18) ,
1; 32 , κx D (cid:19) (60)When both x and x th are much smaller than σ , we usethe expansions given in equation (46) and (54) therebygetting the corresponding approximate formulae T HU ( x th | x , t ) ≈ D ( x th − x ) ( x << σ, and x th << σ ) (61)1for the upcrossing time and T HD ( x th | x , t ) ≈ (cid:114) πκD ( x − x th ) − D ( x − x th ) ( x << σ, and x th << σ ) (62)for the mean downcrossing time. If x is very close to the threshold x th , then the mean upcrossing time islim x → x th T HU ( x th | x , t ) = (cid:114) πDκ e κx th /D erf (cid:18)(cid:114) κD x th (cid:19) ( x th − x ) (63)and mean downcrossing time islim x → x th T HD ( x th | x , t ) = (cid:114) πDκ e κx th /D (cid:18) (erf (cid:18) √ κx th √ D (cid:19) − (cid:19) ( x − x th ) (64)and both of these depend linearly on ( | x − x th | ) whichis the distance between the inititial position of the tip x and the threshold x th .Each of the approximate expressions obtained un-der well defined simple situations expose the inherentasymmetry between the mean times required for hit-ting threshold from below and from above. This is evenmore vividly demonstrated by the following considera-tion. From (49) we extract the mean upcrossing timeform x = 0 to x th = σ and then from x = σ to x th = 2 σ . They are T HU ( σ | , t ) = 0 . κ and T HU (2 σ | σ, t ) = 3 . κ . (65)Similarly, from equation (50) we extract the mean down-crossing time form x = σ to x th = 0 and then from x = 2 σ to x th = σ . They are T HD (0 | σ, t ) = 0 . κ and T HD ( σ | σ, t ) = 0 . κ . (66)Interestingly, the mean upcrossing time T HU ( σ | , t ) toupcross form 0 → σ is smaller than the mean downcross-ing time T HD (0 | σ, t ) to downcross form σ →
0. Contraryto this trend, the mean downcrossing time T HD ( σ | σ, t )to downcross form 2 σ → σ is smaller than the mean up-crossing time T HU (2 σ | σ, t ) to upcross form σ → σ . VI. STATISTICS OF RANDOM EXCURSIONSABOVE A THRESHOLD x th In the last section, we discussed the distributions ofthe duration elapsed, from the initial time, before X ( t )hits a specific threshold or escapes a safe zone for the firsttime. In this section, we will again treat the tip of thefluctuating protrusion as a Brownian particle and ask (i)what is the distribution of the number of hits of the giventhreshold at x th from below (or from above) per unittime, and (ii) what is the distribution of the durations ofits sojourn above (or below) the given threshold [32, 34,35, 61] ? For studying the random excursions made by T fpt 𝜏 - 𝜏 - 𝜏 + 𝜏 + point of upcrossingpoint of downcrossing time ( t ) x(t)x th x*= FIG. 5:
Random excursion of a Brownian particle :
Aschematics of the trajectory of a random particle. First timethe particle hits the threshold x th is denoted by T fpt . Thepoints of upcrossing the threshold x th are marked by opencircles and downcrossing are marked by closed circles. Be-tween and upcrossing and downcrossing, the particle makesa sojourn of duration τ + beyond the threshold x th . τ − de-notes the time between two successive excursions beyond thethreshold x th . the fluctuating length of the protrusion in steady state,we will be using the theory of random excursion above athreshold x th which has been developed for OU processby Stratonovich and is well documented in his book [61].(For a beautiful introduction to random excursion theory,the readers are referred to the book by Brainina [32]). Terminology for excursion theory :
In Fig.5, we haveschematically shown a typical trajectory of a Brownianparticle and have labelled different quantities of interestwhich are studied under the random excursion theory.The threshold x th of interest is marked by a horizontalline in Fig.5. τ + and τ − denote the sojourn times aboveand below, respectively, of the threshold. T fpt is the firstpassage time for hitting the threshold. We shed light onthe statistics of random excursions by computing the (i)mean number of upcrossings of the threshold x th , perunit time and (ii) the distribution of the sojourn times τ + for excursions above the threshold x th .2 A. Mean rate of upcrossings of threshold x th Let x th be the threshold of our interest and x th > x ∗ (see Fig.5). If x denotes the current position of the Brow-nian particle, the time derivative ˙ x ( t ) denotes the rate ofchange of the position of the Brownian particle with time t . Let [ t, t + ∆ t ] be such a small interval that the tra-jectory of the Brownian particle can cross the thresholdnot more than a single time during this interval. Let P c denote the probability that the threshold is crossed onceand P c be the probability of no crossing taking place inthis small interval. As there are only these two possibil-ities, the mean number of crossings taking place in thisinterval is is simply P c . The probability of crossing thethreshold x th in this interval by moving a distance ∆ x with the velocity lying between ˙ x and ˙ x + ∆ ˙ x is given by dP c = p ( x th , ˙ x )∆ x ∆ ˙ x = p ( x th , ˙ x ) ˙ x ∆ t ∆ ˙ x (67)where p ( x th , ˙ x ) is the joint distribution of the Brownianparticle being at x th whose velocity is ˙ x ( t ). We replaced∆ x by ˙ x ∆ t because it is assumed that while moving bya distance ∆ x , the velocity ˙ x ( t ) of the Brownian parti-cle remains constant. Integrating dP c , we get the totalprobability P c of crossing the threshold x th in the inter-val [ t, ∆ t ] i.e, P c = ∆ t (cid:90) ∞ p ( x th , ˙ x ) ˙ xd ˙ x . (68)Since, P c is also the mean number of crossings takingplace in this interval, the mean number density of cross-ing the threshold n c ( x th , t ) per unit time is obtained bydividing P c with ∆ t . So, the expression for n c ( x th , t ) is n c ( x th , t ) = P c ∆ t = (cid:90) ∞ p ( x th , ˙ x ) ˙ x d ˙ x (69)As the system is in steady state, n c ( x th , t ) will be timeindependent quantity. Therefore, we use the stationarydistribution. Moreover, for an OU process, x and ˙ x areuncorrelated. Considering these two points, the expres-sion in equation (69) simplifies to n c ( x th ) = (cid:90) p ss ( x th ) p st ( ˙ x ) ˙ xd ˙ x (70)which, upon evaluation of the integral, as shown in detailin appendix D 1, leads to n c ( x th ) = κ π exp (cid:18) − κx th D (cid:19) . (71)In Fig.6(a), we have plotted n c ( x th ) as a function of x th . In Fig.6(a), we have plotted n c ( x th ) for three dif-ferent values of D by keeping κ fixed and in the Fig.6(a)inset, we have plotted n c ( x th ) for three different valuesof κ by keeping D fixed. As seen in this figure, the rapidfall of n c ( x th ) with increasing x th arises from x th in the Rescaled peak duration ( κ τ + ) n ( κ τ + ) x th =0 x th =20 (b) n c ( x t h ) (a) x th n c ( x t h ) x th κ =10 κ =20 κ =30 D = D = D = FIG. 6:
Quantifying random sojourns : (a) Mean numberdensity n c ( x th ) (equation (71)) as a function of x th is plottedfor fixed κ = 20 and different values of D indicated alongthe curves. Inset - Mean number density n c ( x th ) (equation(71)) as a function of x th is plotted for fixed D = 100 andfor different values of κ indicated along the curves. (b) Meannumber of sojourns n ( κτ + ) per unit time is plotted as functionof the rescaled sojoun time κτ + for two different thresholds- (i) x th = 0 (solid blue line) and (ii) x th = 20 (dashed redline). Value of κ = 9 . × − and D = 8 . × − in (b). factor exp ( − κx th /D ) in Eq.(71). The same exponentialfactor vividly displays the competing roles of κ and D .Larger D tends to enhance random excursions awayfrom the steady state. Therefore, for a given κ , higherthe value of D , higher is the number density n c ( x th )for a given threshold (Fig.6(a)). On the other hand, astronger κ tends to restore the Brownian particle to itsmean position ( x = x ∗ = 0). Keeping D fixed, we sawthat as we increased the value of κ in Fig.6(a) inset, thecurve corrsponding to κ = 30 indicates that the thresh-olds in the neighborhood of the steady state x ∗ = 0 aremore frequently crossed and the crossings become rareras indicated by the rapidly falling curve for higher x th .For smaller values of κ indicates weaker restoring drift.Therefore, the almost flat curve corresponding to κ = 10in Fig.6(a) inset indicates that even the higher thresholdsare crossed frequently. B. Distribution of sojourn times above threshold
We now calculate the mean density of sojourns whoseduration exceed τ + . This will be followed by the eval-uation of the corresponding probability distribution of3sojourn times.The problem is to count all such trajectories which up-cross the threshold x th in the interval [0 , ∆ t ] and do notdowncross it in the next interval [∆ t, τ + ]. All such trajec-tories will give rise to sojourns whose duration exceed τ + .Let all such trajectories be described by the probabilitydensity p + ( x, t ). As these trajectories denote fluctuat-ing position of the Brownian particle in time, probabilitydensity describing them will satisfy the following Fokker-Planck equation ∂p + ( x, t ) ∂t = ∂∂x (cid:20) κx p + ( x, t ) (cid:21) + D ∂ p + ( x, t ) ∂x . (72)The above equation is subjected to the initial condition p + ( x,
0) = 0 for x > x th (73)and boundary conditions p + ( x th , t ) = (cid:40) p ss ( x th ) for 0 ≤ t ≤ ∆ t t < t . (74)The initial condition (73) indicates that only those tra-jectories are considered in which the hypothetical Brow-nian particle is not above the threshold at t = 0. Theboundary condition (74) in the interval [0 , ∆ t ] indicatesthat only those trajectories will contribute which are al-ready at x = x th during the infinitesimal interval [0 , ∆ t ].As the protrusion length is assumed to be in the sta-tionary state, distribution of such trajectories are givenby the stationary distribution p ss ( x th ) given in equation(22). For t > ∆ t , the absorbing boundary condition p + ( x th , t ) = 0 ensures elimination of all those trajecto-ries whose sojourn time above x th would be less than t because of premature downcrossing.Integration of the probability p + ( x, τ + ), that satisfiesthe conditions (73) and (74), over the entire space above x th gives the number of sojourns ∆ n above thresholdwhich begin in [0 , ∆ t ] and do not end by t = τ + , i.e.,∆ n = (cid:90) ∞ x th p + ( x, τ + ) dx. (75)Hence, the mean number density of sojourns above thethreshold x th , each of duration longer than τ + , is givenby n ( τ + ) = lim ∆ t → ∆ n ∆ t (76)The corresponding unnormalised probability density p n ( τ + ) of the duration of sojourn time above thresholdis given by p n ( τ + ) = − dn ( τ + ) dτ + (77)Solving the equation (72), subjected to the set of con-ditions given in equation (73) and (74), is a challenging task. Therefore, we will estimate the sojourn time distri-butions for two special thresholds x th = 0 and x th >> σ only. Special case I : One of the special cases is to put thethreshold at x th = x ∗ = 0. Having X ( t ) > X ( t ) < τ + , abovethreshold in an unit time interval is given by n ( τ + ) = κπ √ e κτ + − n ( τ + ) decreases monotonically with τ + . Us-ing (78) in (77) we get the corresponding sojourn timedistribution as p n ( τ + ) = κπ e κτ + (cid:0) e κτ + − (cid:1) / . (79) Special case II : Another class of important thresholdsare the ones which are rarely visited ( x th >> D/ κ ),which are rarely visited by the Brownian particle. Forsuch high thresholds (( x th >> D/ κ ), the restoring driftis much stronger. This results in excursions of short du-ration as the particle will be able to move by a shorterdistance beyond x th . Therefore, it is safe to assume thata constant drift of magnitude ≈ | κx th | acts on the parti-cle when it is performing excursion beyond the threshold x th i.e, X ( t ) > x th . With this approximation, as shownin appendix D 2, in this special case we get n ( τ + ) = p st ( x th )2 (cid:20)(cid:114) Dπτ + e − ( κx th ) τ + / (2 D ) − ( κx th ) erfc (cid:18) κx th (cid:114) τ + D (cid:19)(cid:21) (80)and p n ( τ + ) = p ss ( x th )2 (cid:114) D π √ τ + ) e − ( κx th ) τ + / (2 D ) . (81)Mean number of sojourns per unit time is plotted as func-tion of the rescaled time κτ + in Fig.6(b) for two differentthresholds - (i) x th = 0 (solid blue line) and (ii) x th = 20(dashed red line). For the curve corresponding to x th = 0(solid blue line in Fig.6(b)), we have used the expressionof n ( τ + ) mentioned in equation (78) and for the one cor-responding to x th = 20 (dashed red line in Fig.6(b)), wehave used the expression of n ( τ + ) mentioned in equation(80). From Fig.6(b) it is very clear that the number ofsojourns of a given duration τ + per unit time beyond alower threshold is more than the number of sojourns be-yond a higher threshold. This trend is expected becauseof the linear nature of the restoring drift which growsstronger with the increasing height of the threshold x th .This prevents excursion of longer duration above higherthresholds. Another inference from Fig.6(b) is that irre-spective of x th , the number of sojourns of sojourn time4 τ + decrease with increasing sojourn time τ + . It is againexpected because the restoring drift will ultimately leadsto the particle reverting back to its mean position at x ∗ = 0. This property leads to lesser number of sojournswith longer sojourn time. VII. EXTREMAL EXCURSIONS
After dealing with the theory of random excursions inthe last section, now we will discuss one more impor-tant aspect of level crossing, namely, the extremal ex-cursions of the protrusion length about its mean length x ∗ = 0 in the steady state. We will look for answers tothe questions like what is the maximum or the minimumlength the protrusion can grow or shorten to and howwide a range does it scan within a finite duration of time[46, 58, 71]. Again mapping the dynamics of protrusiontip onto that of a Brownian particle executing OU pro-cess, we will estimate these important length scales usingthe theories of extremal excusions developed for OU pro-cess in [46]. x*= x ( t ) time ( t ) x max ( t ) x min ( t ) x r a n g e ( t ) FIG. 7:
Extremal excursions :
The trajectory of a Brow-nian particle is denoted by the solid blue line. The evolutionof time dependent maximum x max ( t ) and minimum x min ( t )values attained by the Brownian particle are denoted by thedotted green and dashed red line respectively. The range x range ( t ) spanned is given by the width of the region enclosedby the green dotted line and red dashed line. A. Average maximum, minimum and range
Let x max ( t ) and x min ( t ) be the maximum and the min-imum values that the trajectory of the Brownian particleattains at time t . As shown in Fig.7, both x max ( t ) and x min ( t ) are time dependent random variables.The probability of x max ( t ) being the maximum is de-noted by M max ( t ; ζ | x , t ) = Probability[ x max ( t ) < ζ | x , t ](82)and the probability of x min ( t ) being the minimum is de-noted by M min ( t ; ζ | x , t ) = Probability[ ζ < x min ( t ) | x , t ] . (83) For simplicity, we will keep our discussion limited to thecalculation of statistics of x max . Once we get it for themaximum, getting the results for the minimum x min willbe straight forward.In equation (82), the term Probability[ x max ( t ) <ζ | x , t ] means that the particle which was initially at x ( −∞ < x < ζ ) has not escaped the region [ −∞ , ζ ].In other words, it has not hit the threshold x th = ζ tilltime t . Therefore, M max ( t ; ζ | x , t ) = (1 −H U ( t ; ζ | x , t ))Θ( x − ζ ) . (84)where H U ( t ; ζ | x , t ) is the upcrossing probability for thethreshold ζ which was introduced in equation (40). Theheaviside function Θ( x − ζ ) takes care of the fact thatif initially the particle is located beyond ζ , then it isimpossible to have ζ as the maximum. As the pdf cor-responding to M max ( t ; ζ | x , t ) is p max ( t ; ζ | x , t ) , themean maximum (cid:104) x max ( t ) | x (cid:105) is given by (cid:104) x max ( t ) | x (cid:105) = (cid:90) ∞ ζ p max ( t ; ζ | x , t ) dζ . (85)Rather than having arbitrary values of x , we fix x = 0(= x ∗ ) without loss of generality. This will give usthe average maximum excursion (cid:104) x max ( t ) | x (cid:105) about themean position x ∗ = 0 in steady state. As derived in de-tail in appendix E, it is shown that the average maximum (cid:104) x max ( t ) | (cid:105) goes as (cid:104) x max ( t ) | (cid:105) (cid:39) √ πDt (86)i.e, the (cid:104) x max ( t ) | (cid:105) ∝ √ Dt .Next, let us calculate (cid:104) x min ( t ) | (cid:105) . If initially x = 0,then (cid:104) x min ( t ) | (cid:105) is related to the first downcrossing timeto x th = − ζ . We also know that H D ( t ; − ζ | , t ) = H U ( t ; ζ | , t ) . (87)Therefore, without going through detailed calculationsagain, we can simply write (cid:104) x min ( t ) | (cid:105) = −(cid:104) x max ( t ) | (cid:105) = −√ πDt . (88)The range scanned by the Brownian particle, as shownschematically in Fig.7, is a time dependent random vari-able that depends on both x max ( t ) and x min ( t ). Theaverage width of the range is denoted by (cid:104) x range ( t ) | (cid:105) .It can be evaluated directly by subtracting the averageminimum from the average maximum i.e, (cid:104) x range ( t ) | (cid:105) = (cid:104) x max ( t ) | (cid:105)−(cid:104) x min ( t ) | (cid:105) = 2 √ πDt (89) B. Implications
At first it is surprising that all the three quantities (cid:104) x max ( t ) | (cid:105) , (cid:104) x min ( t ) | (cid:105) and (cid:104) x range ( t ) | (cid:105) (see equation(86),(88) and (89)) characterizing the extremal excur-sions are proportional to √ Dt just like the root mean5square displacement in pure diffusion. The most inter-esting feature of these expressions for the three quantitiesis that none of these depend on κ at all. Since it is solelybecause of diffusion that the particle tends to move awayfrom the mean position and explore extremes, all thesethree time-dependent quantities vary with time as √ Dt ,the hall mark of diffusion. VIII. EUKARYOTIC FLAGELLUM: A CASESTUDY
Internal structure :
Eukaryotic flagellum is a mem-brane bound organelle that projects out of the cell body.The primary structural scaffold which forms the flagellumis the axoneme [72]. Axoneme is a microtubule (MT)-based structure consisting of nine MT doublets and 2single MTs (see Fig.8). The doublets are arranged ina cylindrical fashion with the two singlets at the cen-ter. Absence of mRNAs inside the flagellum indicatesthat the flagellum imports its structural proteins (nowonwards, loosely, referred to as “precursor”) from the cellbody. The flagellar precursors are synthesized and anddegraded in the pool which is situated at the flagellarbase [72].
Intraflagellar transport :
An active transportmechanism known as intraflagellar transport (IFT) isresponsible for transferring precursors into and out ofthe flagellum [74–76]. IFT trains consist of IFT proteinswhich are arranged in linear arrays just like the bogies ofa train (see Fig.8). These are pulled by molecular motorswhich walk on the MT doublets. The precursors hitchikeon these trains to get a ride inside the flagellum. Oneof the MTs of the doublet (B-MT) is exlusively used forthe anterograde trip of the IFT trains to the tip and theother one (A-MT) is used for the retrograde trip from tipto the base [77]. The IFT trains are driven towards theflagellar tip by the kinesin motors. At the tip, the trainsget ‘remodelled’, kinesins are ‘disengaged’ and the dyneinmotors get ‘activated’. Thereafter, they are pulled backto the base by the dynein motors. Due to the use ofseparate MTs, anterograde and retrograde trains nevercollide.
Length control :
A flagellum elongates by incor-porating the precursors,which are brought by the IFTtrains, at its tip . Because of the turnover of tubulinsat the flagellar tip, the flagellum shortens by removal ofprecursors [11, 38, 72, 78]. The overall assembly rate de-creases with the increasing flagellar length whereas thedisassembly rate remains constant throughout the pro-cess [11, 38, 72, 78]. A steady state length emerges bythe balance of elongation and retraction. According tothe differential loading model, the amount of precursorsloaded onto an IFT train decreases with the increasinglength of the flagellum and this is responsible for theoverall decrease in the assembly rate with the growinglength of the flagellum. The cell senses the flagellarlength with a “time of flight” mechanism [11, 18, 78].
Tubulin dimer IFT train Kinesin Dynein
Longitudinal section of fl agellumChlamydomonas Reinhardtii(Green algae) C r o ss s e c t i o n o f fl a g e ll u m FlagellumPrecursorpool Tip
FIG. 8: A glimpse into the internal structure of flagellumand elements of intraflagellar transport.
Suppose, the timer molecule, which is an integral partof the IFT trains, enters the flagellum in a particularchemical (or, conformational) state. It can switch irre-versibly to another state at a certain rate. The state ofthe timer molecule, which finishes the roundtrip insidethe flagellum, gives a feedback to the cell as to the cur-rent flagellar length. Longer the flagellum, longer thetime of flight and higher is the probability of flipping ofthe state of the timer. If the timer returns to the basewith its state unchanged, then it implies that precursorshave to be loaded on to next IFT train, but otherwiseno cargo is loaded onto the next IFT train that starts itsround trip inside the flagellum.
Model :
Here we summarize the key points of astochastic model that we have developed very recentlyfor studying flagellar length control (see [18] for the de-tails). The flagellum is represented as a pair of antipar-allel lattices, where each lattice represents a MT of thedoublet. IFT particles pulled by the motors are repre-sented by self driven particles which obey exclusion prin-ciple and jump to the neighboring sites in a stochasticmanner with a certain hopping rate. The flux of the IFTparticles inside the flagellum is J , their number densityon the lattice is ρ and the effective velocity with whichthey move is v . Each IFT particle can be either emptyor loaded with a single lattice unit which can elongatethe model flagellum (pair of lattices) at the tip by a sin-gle unit. Whether the IFT train is loaded or not witha lattice unit depends on the state of the timer and theamount of precursor in the pool. If initially the timer isin a given state, the probability of finding the timer inthe same state after time t is given by exp ( − kt ) where k is rate of flipping of the state of the timer. So, if theflagellar length is L ( t ) at time t , the probability of thetimer returning to the base (after the roundtrip insideflagellum) without flipping state is e − kt tof where t tof isthe time of flight and is given by t tof = 2 L ( t ) /v . So faras the precursor pool is concerned, if the maximum ca-pacity of pool is N max and by some mechanism the cellmaintains a steady amount of precursors N ss in the pool6then the probability of loading of precursors onto an IFTtrain is proportional to N ss /N max . Therefore the flux offull trains reaching the flagellar tip is J full = N ss N max e − kL ( t ) /v J (90)The precursor loaded on the IFT train can elongate theflagellum with probability Ω e . So the overall rate of as-sembly is given by J Ω e e − kL ( t ) /v . (91)Due to ongoing turnover, if both the sites at the tip arenot occupied by any IFT trains, the dimer (i.e the pre-cursor) dissociates with rate Γ r . Therefore, the overalldisassembly rate is given by(1 − ρ ) Γ r (92)The stochastic time-evolution of the flagellar length canbe modelled by a a master equation (1) with the corre-sponding rates r + ( (cid:96) ) = Ae − C(cid:96) and r − ( (cid:96) ) = B (93)where A = N ss N max J Ω e , B = (1 − ρ ) Γ r & C = 2 k/v . (94) A. Parameter estimation for eukaryotic flagellum
Now we estimate the model parameters relevant forthe steady state length fluctuations of the flagellum of
Chlamydomonas reinhardtii . We chose this species be-cause the flagellum of this particular species of green al-gae is most commonly used in experimental studies oflength control [11, 38, 78]. Based on our earlier expe-rience of analysis of experimental data for [18], we fix A = 809 . − , B = 28 .
21 min − and C = 0 . (cid:104) (cid:96) ( t ) (cid:105) using equation (9)and multiply (cid:104) (cid:96) ( t ) (cid:105) with 8 nm for converting it into ac-tual length; 8 nm being the length of a single MT dimer.Plugging the values of A , B and C in the expressions inequation (33), we get the corresponding values of κ − = ( BC ) − = 16 . D = 2 B = 56 .
42 min − (96)which are the two essential parameters of the OU process.Note that the mean length in steady state for the flag-ellum is (cid:104) (cid:96) ss (cid:105) = 1 C (cid:96)og (cid:18) AB (cid:19) = 12 . µm, (97)which is, indeed, the typical length of a flagellum of Chlamydomonas reinhardtii , and σ = (cid:114) D κ = (cid:114) C = 0 . µm (98) B. Level-crossing statistics for eukaryotic flagellum
First we estimate the mean hitting times for a flagel-lum of
Chlamydomonas reinherdtii using the expressionsof hitting times (65,66) derived in section V. The meanhitting time for important thresholds is listed in Fig.9(b)and the timescales are presented in measurable units (inminutes). The mean time taken by the flagellar length tohit the threshold (cid:104) (cid:96) ss (cid:105) + σ is around 10 minutes whereasthe mean time to regain the mean length (cid:104) (cid:96) ss (cid:105) is around15 minutes. Notice the asymmetry in mean times in hit-ting equidistant thresholds in opposite directions.Moreover, it takes around 1 hour, on the average, togrow from (cid:104) (cid:96) ss (cid:105) + σ to (cid:104) (cid:96) ss (cid:105) + 2 σ by fluctuations whereasit can shorten to (cid:104) (cid:96) ss (cid:105) + σ from (cid:104) (cid:96) ss (cid:105) + 2 σ in around 8minutes, on the average. From the estimates of hittingtime, it is concluded that upcrossing from (cid:104) (cid:96) ss (cid:105) + σ to (cid:104) (cid:96) ss (cid:105) +2 σ is more difficult than downcrossing from (cid:104) (cid:96) ss (cid:105) +2 σ to (cid:104) (cid:96) ss (cid:105) + σ whereas downcrossing from (cid:104) (cid:96) ss (cid:105) + σ to (cid:104) (cid:96) ss (cid:105) is more challenging than upcrossing from (cid:104) (cid:96) ss (cid:105) to (cid:104) (cid:96) ss (cid:105) + σ .Next we estimate the number density of thresholdcrossings and the sojourn times above a threshold, whichare statistical properties of random excursions discusedin section VI. The mean number n c ( x th ) of times a giventhreshold x th is crossed per minute (see expression (71))is plotted as a function of x th − (cid:104) (cid:96) ss (cid:105) in Fig.9(c1). Thenumber of crossings per minute of the particular thresh-old x th − (cid:104) (cid:96) ss (cid:105) = σ is also marked in Fig.9(c1). Again,the decreasing trend of the curve reconfirms the fact thatupcrossing the higher thresholds is more difficult andrare. We have also estimated the number of sojourns n ( τ + ) above a threshold x th per minute whose durationexceed τ + (in minutes). We get the estimate for threespecial thresholds (i) x th = 0, (ii) x th = (cid:104) (cid:96) ss (cid:105) + σ and (iii) x th = (cid:104) (cid:96) ss (cid:105) + 2 σ . Using the expressions (78) and (80), weplot n ( τ + ) as a function of τ + for these three thresholds.For all the three threshold, n ( τ + ) decrease with increas-ing τ + indicating that number of sojourns with sojourntime longer than τ + decrease with increasing τ + .Finally, we address the questions on the maximum andminimum lengths a flagellum can attain during its life-time merely by fluctuations beginning from its steady-state. For this purpose we get the relevant estimates byusing the expression (86),(88) and (89). Chlamydomonasreinhardtii assembles its flagellum at the beginning of G1phase of the cell cylce and starts to disassemble at the endof this phase when the cell prepares itself for cell division.G1 phase lasts for 10-12 hours. About its steady statelength (cid:104) (cid:96) ss (cid:105) ≈ µm , the maximum and minimum aver-age length by which the flagellum can grow or shortenduring its lifetime of 10 hours is 2.5 µ m i.e, it can growto ≈ . µ m and shorten to ≈ . µ m due to length fluc-tuation in 10 hours. The range scanned by the flagellartip during its lifetime is, thus, about 5 µ m.7 μ m12.3 + 0.17 μ m12.3 + 0.34 μ m12.3 - 0.17 μ m12.3 - 0.34 μ m 〈 ℓ ss 〉〈 ℓ ss 〉 + σ 〈 ℓ ss 〉 + 2 σ 〈 ℓ ss 〉 - σ 〈 ℓ ss 〉 - 2 σ
50 100 15024681012
50 100 15024681012
Time (in minutes) F l a g e ll a r l e n g t h ( i n μ m ) (a) Evolution of mean fl agellar length (b) Upcrossing and downcrossing times for special thresholds x th - 〈 ℓ ss 〉 (in μ m) x t h - 〈 ℓ ss 〉 = σ n c ( x t h ) ( p e r m i n u t e ) n ( τ + ) ( p e r m i n u t e ) τ + (in minute) x t h = 〈 ℓ ss 〉 + σ = . + . μ m x t h = 〈 ℓ ss 〉 + σ = . + . μ m x t h = 〈 ℓ ss 〉 = . μ m (c) Statistics of random excursions (c1) (c2) Time (in hours) F l a g e ll a r l e n g t h ( i n μ m ) A v e r a g e m a x i m u m l e n g t h A v e r a g e m i n i m u m l e n g t h R a n g e s c a nn e d a b o u t t h e m e a n l e n g t h ( i n μ m ) (d) Statistics of extreme excursions (d1) (d2) Time (in hours)
FIG. 9:
Length fluctuations of eukaryotic flagellum : (a) Evolution of the mean fagellar length which we obtainedby solving the rate equation 9 by using the values A = 809 . min − , B = 28 . min − and C = 0 . µ m. (b) Hitting times for special thresholds in flagellum. Levels are indicated by the horizontal. Tip of the arrows indicatethe terminating point and their base (with a circle) indicate the initiating point. The mean upcrossing and downcrossing timesare denoted besides these arrow which indicate the transition. (c) Statistics of random excursions. (c1) Number of upcrossings n ( x th ) per unit minute as a function of threshold x th . (c2) Number of sojourns per unit time above a given threshold (indicatedalong the lines) whose duration exceed τ + minutes. (d) Statistics of extreme excursions - (d1) Average maximum (denoted bydotted green line) and minimum flagellar length (dashed red line) about its mean length as a function of time. (d2) Averagerange scanned by the flagellar tip as a function of time. For (b-d), κ = 0 .
061 min − and D = 56 .
42 min − IX. SUMMARY AND CONCLUSION
In a classic essay, titled “on being the right size”, J.B.S.Haldane [2] first analyzed the physical reasons that wouldexplain why “for every type of animal there is a conve-nient size”. Haldane focused his analysis on the size ofwhole organisms. The mechanisms that ensure the “con-venient” size of a cell [7] and sub-cellular structures [5–7]have become a very active field of research in recent years.There are three different kinds of questions that needto be addressed:(I) The relations between the size and various other pa-rameters that characterize the structure or dynamics ofthe organism; these observations have been presentedmostly in the form of scaling relations that are usuallyreferred to allometric relations.(II) The allometric relations are explained in terms ofconstraints imposed by the laws of physics and chemistry;these are, for example, mechanical strength or chemicalreaction rate, etc. Having understood the causes of se-lecting the specific convenient size, one also needs to un-derstand functional consequences of any deviation fromthat convenient size caused, possibly, by mutations.(III) Having discovered the allometric laws satisfied bythe size and the physico-chemical origin of these laws,the next question is how does an organism grow up toa given size, stop further growth and maintain it subse-quently.It is the type (III) questions that we have addressed ina recent paper [18]. However, instead of a multi-cellularorganism or a single cell, we have considered the size ofsubcellular structures. Long protrusions and membrane-bound organelles, that appear as cell appendages, areprominent among the sub-cellular structures. Except fora few papers, including ref.[18], all the works on lengthcontrol of cell protrusions so far have investigated thetime-dependence of the mean lengths. Moreover, thosehave mostly assumed either length-dependent growth orlength-dependent shrinkage of the respective protrusions.Going beyond the mean length, in this paper we haveinvestigated various aspects of the fluctuations of thelengths of the cell protrusions in the steady-state. Us-ing the formalisms of stochastic processes, we have firstmapped the time evolution of the tip of a dynamic pro-trusion on an Ornstein-Uhlenbeck process. The mappingis very general and is applicable for all the three classesof protrusions listed in Table I. Then, we derive ana-lytical expressions for several statistical characteristicsof the fluctuations using the techniques of level crossingpioneered by Rice [34], and pedagogically presented byStratonovich [61], Brainina [32], Masoliver [46] and oth-ers. The results could be segregated into three groupsand summarized in table II.For a specific case study, we have considered eukaryoticfagellum (also called a cilium). We have first extracted the rate parameters from the experimental data availablein the literature. Then, plugging these rates into the an-alytical expressions that we have dereived in this paper,
Quantities Symbol FormulaStatistics of passage times
Mean exit time 𝓣 ε ( x U , x L | x t ) Equation (45)Mean upcrossing time 𝓣 HU ( x th | x t ) Equation (49)Mean downcrossing time 𝓣 HD ( x th | x t ) Equation (50) Statistics of random excursion
Number density of upcrossings for the threshold x th n c Equation (70)Number density of sojourns above a threshold with sojourn time 𝜏 + n ( 𝜏 + ) Equation (78)Equation (80)Distribution of sojourn time above a threshold x th p n ( 𝜏 + ) Equation (79)Equation (81)
Statistics of extremal excursion
Average maximum value attained about the mean position as a function of time . 〈 x max ( t )|0 〉 Equation (86)Average maximum value attained about the mean position as a function of time . 〈 x min ( t )|0 〉 Equation (88)Average range spanned about the mean position as a function of time . 〈 x range ( t )|0 〉 Equation (89)
TABLE II: List of important quantities derived in this paperwhich describe different aspects of level crossing for protru-sions and filaments with fluctuating length. we have plotted the statistical properties of the fluctua-tions of the lengths of the flagellum in the steady-statein Fig.9. As byproduct of this analysis we also get roughestimates of the typical length scales and time scales in-volved. Based on these estimates, we conclude that ourpredictions can be tested by analyzing a series of ky-mographs capturing the temporal evolution of length inwild type and mutant cells. We also believe that studiesof fluctuations of the lengths of sensory cilia, in particu-lar, will shed light on ‘fading’ of chemical signals just asRice’s pioneeing work on level crossing was motivated byfading of radio signals [34].
Acknowledgement :
SP acknowledges support fromIIT Kanpur through a teaching assistantship. D.C. ac-knowledges support from SERB (India) through a J.C.Bose National Fellowship.9
Appendix A: Equivalence of Fokker-Planck and stochastic differential (Langevin) equation
The stochastic differential equation is dy = R − ( y ) dt + (cid:112) R + ( y ) √ N dW ( t ) (A1)Let f [ y ] be an arbitrary function of y . Expanding f [ y ] in second order: df [ y ] = f [ y + dy ] − f [ y ]= f [ y ] + f (cid:48) [ y ] dy + 12 f (cid:48)(cid:48) [ y ][ dy ] − f [ y ]= f (cid:48) [ y ] (cid:18) R − ( y ) dt + (cid:112) R + ( y ) √ N dW ( t ) (cid:19) + 12 f (cid:48)(cid:48) [ y ] (cid:18) R − ( y ) dt + (cid:112) R + ( y ) √ N dW ( t ) (cid:19) N ote : Neglecting the terms containing ( dt ) and dtdW ( t ) (cid:39) f (cid:48) [ y ] (cid:18) R − ( Y ) dt + (cid:112) R + ( y ) √ N dW ( t ) (cid:19) + 12 f (cid:48)(cid:48) [ y ] (cid:18) R + ( y ) N ( dW ( t )) (cid:19) N ote : Using the relation ( dW ( t )) = dt = (cid:18) R − ( y ) f (cid:48) [ y ] + 12 N R + ( y ) f (cid:48)(cid:48) [ y ] (cid:19) dt + (cid:18) (cid:112) R + ( y ) √ N f [ y ] (cid:19) dW ( t ) (A2)Let us consider the time development of the arbitrary function f [ y ( t )]. If we take the average of df [ y ] over realizationsand divide by dt , we get d (cid:104) f [ y ] (cid:105) dt = ddt (cid:28)(cid:18) R − ( y ) f (cid:48) [ y ] + 12 N R + ( y ) f (cid:48)(cid:48) [ y ] (cid:19) dt + (cid:18) (cid:112) R + ( y ) √ N f [ y ] (cid:19) dW ( t ) (cid:29) = (cid:28) R − ( y ) ∂f [ y ] ∂y + 12 N R + ( y ) ∂ f [ y ] ∂y (cid:29) (A3)Now y evolves according to the conditional density p ( y, t | y , t ). The left hand side of the above equation (A3) issimply given by d (cid:104) f [ y ] (cid:105) dt = ddt (cid:90) dY f ( y ) p ( y, t | y , t ) = (cid:90) dyf ( y ) ∂p ( y, t | y , t ) ∂t (A4)(Because f ( y ) is not a function of t , therefore we can take it out. On the other hand, p ( y, t ) is a function of both y and t . Therefore, there will be a partial derivative. This is how Gardiner proceeds from here.)Now going to the right hand side term of equation (A3) (cid:28) R − ( y ) ∂f [ y ] ∂y + 12 N R + ( y ) ∂ f [ y ] ∂y (cid:29) = (cid:90) dy (cid:18) R − ( y ) ∂f [ y ] ∂y + 12 N R + ( y ) ∂ f [ y ] ∂y (cid:19) p ( y, t | y , t )= (cid:90) dyf [ y ] (cid:18) − ∂R − ( y ) p ( y, t | y , t ) ∂y + 12 N ∂ R + ( y ) p ( y, t | y , t ) ∂y (cid:19) (A5)Equating the last expressions of equation (A4) and (A5) we get (cid:90) dY f ( y ) ∂p ( y, t | y , t ) ∂t = (cid:90) dY f [ y ] (cid:18) − ∂R − ( y ) p ( y, t | y , t ) ∂y + 12 N ∂ R + ( y ) p ( y, t | y , t ) ∂y (cid:19) . (A6)As f [ y ] is arbitrary, we can simply write the following ∂p ( y, t | y , t ) ∂t = − ∂R − ( y ( t )) p ( Y, t | y , t ) ∂y + 12 N ∂ R + ( y ( t )) p ( Y, t | y , t ) ∂y (A7)Therefore, we can say that for the the following SDE dy = R − ( y ) dt + (cid:112) R + ( y ) √ N dW ( t ) (A8)0the corresponding Fokker-Planck equation is given by ∂p ( y, t ) ∂t = − ∂R − ( y ) p ( y, t ) ∂y + 12 N ∂ R + ( y ) p ( y, t ) ∂y (A9)and vice versa. Appendix B: Mapping into Ornstein-Uhlenbeck procees
We have y − y ∗ = x √ N ⇒ y = y ∗ + x √ N (B1)Substituting this in equation (11) we get d ( y ∗ + x √ N ) = R − ( y ∗ + x √ N ) dt + 1 √ N (cid:114) R + ( y ∗ + x √ N ) dW ( t ) ⇒ dx √ N = (cid:18) R − ( y ∗ ) + x √ N R (cid:48) + ( y ∗ ) (cid:19) dt + 1 √ N (cid:115)(cid:18) R + ( y ∗ ) + x √ N R (cid:48) + ( y ∗ ) (cid:19) dW ( t ) (B2)Using the defination of R − ( y ) (equation (6)) and equation (10), we get R − ( y ∗ ) = 0. Neglecting the term R (cid:48) + ( y ∗ )inside the square root we get ⇒ dx √ N = x √ N R (cid:48) + ( y ∗ ) dt + 1 √ N (cid:112) R + ( y ∗ ) dW ( t ) ⇒ dx = R (cid:48) + ( y ∗ ) x dt + (cid:112) R + ( y ∗ ) dW ( t ) (B3)which is the transformed stochastic differential equation(13) we wanted to derive from equation (11). Appendix C: Statistics of passage times1. Mean exit time
The escape probability E ( t ; x U , x L | x , t ) is introducedin equation (39). For our convinience, we denote it simplyby E ( x , t ) . Integrating the backward FPE (38) withrespect to final position x (as done while defining theescape probability in equation (38)) it can be checkedthat the escape probability E ( x , t ) obeys the backwardFPE i.e, ∂ E ( x , t ) ∂t = − κx ∂ [ E ( x , t ) ∂x + D ∂ E ( x , t ) ∂x (C1) subjected to the initial condition E ( x , t ) = 0 (C2)and the following boundary conditions E ( x U , t ) = 1 and E ( x L , t ) = 1 (C3)The initial condition (equation (C2)) indicates that ini-tially x lies between x U and x L and the boundary con-dition (equation (C3)) indicates whenever the length hitseither of the thresholds, it successsfully exits the zone. Using the Laplace transform˜ E ( x , s ) = (cid:90) ∞ e − st E ( x , t ) dt (C4)the backward FPE (equation (C1)) gets converted to anordinary differential equation − κx d ˜ E ( x , s ) dx + D d ˜ E ( x , s ) dx = s ˜ E ( x , s ) − E ( x L , s ) = ˜ E ( x U , s ) = 1 /s (C6)Now onwards, for our convinience, we will denote the cor-responding pdf p E ( t, x U , x L | x , t ) by a simpler notation p E ( x , t ). The moments of the exit time are given by T n E ( x U , x L | x , t ) = (cid:90) ∞ t n p E ( x , t ) dt (C7)where the zeroth moment corresponding to n = 0 is T E = 1 and the first moment corresponding to n = 1is the mean exit time T E ( x U , x L | x , t ). Let us denotethe moments of exit time T n E ( x U , x L | x , t ) simply by T n E ( x ). If ˜ p E ( x , s ) is the Laplace transform of the pdf p E ( x, t ), the moments of escape time are given by T n E ( x ) = ( − n ∂ n ∂s n ˜ p E ( x , s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 (C8)1Therefore, ˜ p E ( x , s ) can be expanded and its series formis given by the Laplace transform ˜ p E ( x , s ) can be writtenas a power series of the Laplace variable s ˜ p E ( x , s ) = ∞ (cid:88) n =0 ( − n n ! T n E ( x ) s n (C9)provided all the moments of the escape time exists. p E ( x , t ) is obtained from the probability E ( x , t ) bytaking derivate with respect to tp E ( x , t ) = ∂ E ( x , t ) ∂t . (C10)Taking into account the fact that E ( x ,
0) = 0 which indi-cates that initially it is impossible to escape the safe zone,it can be shown that the relation between the Laplacetransforms ˜ E ( x , s ) and ˜ p E ( x , s ) is˜ p E ( x , s ) = s ˜ E ( x , s ) (C11)which is obtained by taking the Laplace transforms of theterms on both side of the equation (C10). From equation (C9) and(C11), we can see that the˜ E ( x , s ) = 1 s ∞ (cid:88) n =0 ( − n n ! T n E ( x ) s n = 1 s (cid:18) − s T E ( x ) + ∞ (cid:88) n =2 ( − n n ! T n E ( x ) s n (cid:19) (C12)On rearranging the above equation1 s − ˜ E ( x , s ) = T E ( x ) − ∞ (cid:88) n =2 ( − n n ! T n E ( x ) s n − (C13)and taking the limit s → s → (cid:20) s − ˜ E ( x , s ) (cid:21) = T E ( x ) . (C14)Plugging this relation mentioned in equation (C14) intothe backward Fokker-Planck equation (C1), we getlim s → (cid:26) − κx ddx (cid:20) s − T E ( x ) (cid:21) + D d dx (cid:20) s − T E ( x ) (cid:21)(cid:27) = lim s → (cid:26) s (cid:20) s − T E ( x ) (cid:21) − (cid:27) (C15)which simplifies to − κx d T E ( x ) dx + D d T E ( x ) dx = − . (C16)On rearranging the corresponding boundary condition(C6) and taking limitslim s → (cid:20) s − ˜ E ( x U ) (cid:21) = lim s → (cid:20) s − ˜ E ( x L ) (cid:21) = 0 (C17)we get the boundary condition T E ( x U ) = T E ( x L ) = 0 . (C18)for the ordinary differential equation (C16).
2. Mean hitting time
By integrating the backward FPE (38) with respectto the final position and using the defination of hittingprobability given in equation (40) or (41), we can showthat H ( t ; x th | x , t ) (simply written as H ( x , t )) obeysthe backward FPE: ∂ H ( x , t ) ∂t = − κx ∂ H ( x , t ) ∂x + D ∂ H ( x , t ) ∂x (C19) with initial condition H ( x ,
0) = 0 (C20)and boundary condition given by H ( x th , t ) = 1 . (C21)We rescale t by multiplying it with κ . So, replacingthe t with the rescaled time t r which is given by t r = κt . (C22)we transform the partial differential equation ∂ H ( x , t r ) ∂t r = − x ∂ H ( x , t r ) ∂x + D κ ∂ H ( x , t r ) ∂x (C23)The laplace transform of the hitting probability is givenby ˜ H ( x , s ) = (cid:90) ∞ e − st r H ( x , t r ) dt r (C24)and using this Laplace transform, we convert the partialdifferential equation (equation (C23)) into an ordinarydifferential equation given by − x d ˜ H ( x , s ) dx + D κ d ˜ H ( x , s ) dx = s ˜ H ( x , s ) (C25)2which is subjected to the boundary condition˜ H ( x th , s ) = 1 s . (C26)Using the change of variables z = κD x (C27)one can check that equation (C25) can be recasted to thefollowing second order ODE z d ˜ H ( z, s ) dz + (cid:18) − z (cid:19) d ˜ H ( z, s ) dz − s H ( z, s ) = 0 (C28)whose general solution is a linear combination of Kummerfunctions. Hence the general solution of the equation(C25) is given by˜ H ( x , s ) = C F ( s/ , / , κx /D )+ C U ( s/ , / , κx /D ) (C29)Now let us have a look at the asymptotic behaviour ofthe Kummer functions. In the limit x → ±∞ lim x →±∞ F ( s/ , / , κx /D ) == Γ(1 / s/ x s − e κx /D [1 + O (1 /x )] → ∞ (C30)so, it can serve to be the solution when x is bounded i.e, x < x th . On the other hand,lim x →±∞ U ( s/ , / , κx /D ) = | x | − s [1 + O (1 /x )] → x remains unbounded, i.e, x > x th . C and C can be evaluated using the boundary condi-tion given by equation (C26) and collectively, the uniquesolution of equation (C25) is given by˜ H ( x , s ) = F ( s/ , / ,κx /D ) sF ( s/ , / ,κx th /D ) when | x | < | x th | U ( s/ , / ,κx /D ) sU ( s/ , / ,κx th /D ) when | x | > | x th | (C32)Using the arguments used to derive the relation statedin equation (C14) , we can arrive at the following T r H ( x th | x , t ) = lim s → (cid:20) s − ˜ H ( x ) (cid:21) (C33)where T r H ( x th | x , t ) ( simply denoted by T r H ( x )) is therescaled mean hitting time. For x < x th , the rescaled mean hitting (upcrossing)time T r HU ( x ) for hitting the threshold is T r H ( x ) = T r HU ( x ) = lim s → (cid:20) s − F ( s/ , / , κx /D ) sF ( s/ , / , κx th /D ) (cid:21) . (C34)Plugging the following relation in the above equation F ( s/ , / , κx /D )= 1 + s (cid:90) κx /D F (1 , / , z ) dz + O [ s ] (C35)we get T HU ( x ) = 1 κ T r HU ( x ) = 1 κ (cid:90) κx th /Dκx /D F (1 , / , z ) dz = √ πκ (cid:90) √ κ | x th | / √ D √ κ | x | / √ D e y erf( y ) dy (C36)For x > x th , the rescaled mean hitting (downcrossing) T r HD ( x ) time for hitting the threshold is T r H ( x ) = T r HD ( x ) = lim s → (cid:20) s − U ( s/ , / , κx /D ) sU ( s/ , / , κx th /D ) (cid:21) . (C37)Plugging the following relation in the above equation U ( s/ , / , κx /D ) =1 − s (cid:20) ψ (1 /
2) + (cid:90) κx /D U (1 , / , z ) dz + O [ s ] (cid:21) (C38) we get T HD ( x ) = 1 κ T r HD ( x ) = 12 κ (cid:90) κx /Dκx th /D U (1 , / , z ) dz = √ πκ (cid:90) √ κ | x | / √ D √ κ | x th | / √ D e y erfc( y ) dy (C39) Appendix D: Statistics of random excursions1. Mean density of crossing a threshold
Here we perform the integration in equation (70) toget the expression of mean density of upcrossings whichwe directly wrote in equation (71).The steady state distribution of protrusion length( X ( t )) is a Gaussian distribution1 σ x √ π e − x / (2 σ x ) (D1)with mean µ x given by µ x = x ∗ = 0 (D2)3and standard deviation σ x given by σ x = (cid:114) D κ (D3)Since x follows a Gaussian distribution, then its deriva-tive ˙ x also obeys a Gaussian distribution with mean µ ˙ x = 0 and standard deviation σ ˙ x given by σ x = −C (cid:48)(cid:48) ( τ ) | τ =0 = − ddτ (cid:18) − D κ e − κτ (cid:19) τ =0 = Dκ (D4)For integrating the expression in equation (70), we usethe following result (cid:90) ∞ ˙ xp ( ˙ x ) d ˙ x = (cid:90) ∞ ˙ x σ ˙ x √ π e − ˙ x / (2 σ x ) d ˙ x = σ ˙ x √ π . (D5)Using this, we get the expression for mean density n c ( x th )as n c ( x th ) = p ss ( x th ) (cid:90) ∞ ˙ xp ( ˙ x ) d ˙ x = 12 π σ ˙ x σ x e − x / (2 σ x ) = κ π e − κx th /D (D6)
2. Statistics of sojourns above a threhold
For obtaining the expression for n ( τ + ), we need theexpression for p + ( x, t ) and the following limitlim t → p + ( x, t )∆ t (D7)For convenience , let us introduce the following function w + ( x, t ) w + ( x, t ) = 1 p st ( x ) lim t → p + ( x, t )∆ t (D8)Just as p + ( x, t ) satisfies the Fokker-Planck equation (72),this new function w + ( x, t ) also satisfies the followingFokker-Planck equation ∂w + ( x, t ) ∂t = ∂∂x (cid:20) κx w + ( x, t ) (cid:21) + D ∂ w + ( x, t ) ∂x (D9)and from the initial and boundary conditions (equation(73) and (74) respectively) we can infer that the FPE(equation (D9)) is subjected to the following conditions w + ( x, t ) = 0 for t < ,w + ( x th , t ) = δ ( t ) . (D10)In terms of w + ( x, t ), n ( τ + ) is given by n ( τ + ) = p st (cid:90) ∞ x th w + ( x, τ ) dx (D11) The partial differential equation (D9) will be solved byusing the Carlson-Laplace transform. The transform of w + ( x, t ) is given by˜ w + ( x, s ) = s (cid:90) ∞ e − st w + ( x, t ) dt (D12)The Fokker-Planck equation (D9) in terms of ˜ w + ( x, s )is s ˜ w + ( x, s ) = ∂∂x (cid:20) κx ˜ w + ( x, s ) (cid:21) + D ∂ ˜ w + ( x, s ) ∂x (D13)and the boundary condition takes the form˜ w + ( x th , s ) = s (D14)Using this boundary condition and the fact that the cor-responding flux vanishes as x → ∞ , the Fokker-Planckequation (D13) can be integrated to (cid:90) ∞ x th ˜ w + ( x, s ) dx = − s (cid:20) D ∂ ˜ w + ( x th , s ) ∂x + κx th ˜ w + ( x th , s ) (cid:21) (D15)Rearranging Eq.(D13), we get ∂ ˜ w + ( x, s ) ∂x + (cid:18) κD (cid:19) x ∂ ˜ w + ( x, s ) ∂x + (cid:18) κD (cid:19)(cid:18) − sκ (cid:19) ˜ w + ( x, s ) = 0(D16)which, upon change of variable from x to z defined by z = (cid:114) κD x, (D17)gets transformed to ∂ ˜ w + ( z, s ) ∂z + z ∂ ˜ w + ( z, s ) ∂z + (cid:18) − sκ (cid:19) ˜ w + ( z, s ) = 0 . (D18)The general solution for the above equation has the form˜ w + ( x, s ) = C e − z / D − s/k ( z ) + C e − z / D − s/k ( − z )(D19)where D − s/k ( y ) is the parabolic cylindrical function and C and C are arbitrary constants fixed by the boundaryconditions. As the first term vanishes as z → ∞ whereasthe second does not, it qualifies as the physically allowedsolution. Imposing the boundary condition (D14) at x = x th we get˜ w + ( x th , s ) = s = C e − ( κx th ) / (2 D ) D − s/k (cid:18)(cid:114) κD x th (cid:19) (D20)which gives the expression for C that we use to writethe solution˜ w + ( x, s ) = s exp (cid:18) x th − x D (cid:19) D − s/k ( (cid:113) κD x ) D − s/k ( (cid:113) κD x th ) (D21)4The Carlson-Laplace transform of n ( τ ) is denoted by˜ n ( s ) and the expression of ˜ n ( s ) using equation (D8),(D11) and (D15), we get˜ n ( s ) = − p st ( x ) s (cid:20) D ∂ ˜ w + ( x th , s ) ∂x + κx th ˜ w + ( x th , s ) (cid:21) (D22) Substituting the expression for ˜ w + ( x, s ) in equation(D22) and after some rearrangement we get˜ n ( x, s )= − p st ( x th ) × (cid:20) D D − s/k ( (cid:113) κD ) x th ) exp (cid:18) κ ( x th − x th )2 D (cid:19)(cid:18)(cid:114) κD (cid:19)(cid:26) − (cid:114) κD D − s/k (cid:18)(cid:114) κD x th (cid:19) + D (cid:48)− s/k (cid:18)(cid:114) κD x th (cid:19)(cid:27) + κx th (cid:21) ( Using the identity D (cid:48) ν ( z ) − z D ν ( z ) + D ν +1 ( z ) = 0 )= − p st ( x th ) (cid:20) D D − s/k ( (cid:113) κD ) x th ) (cid:18)(cid:114) κD (cid:19)(cid:26) −D − s/k +1 (cid:18)(cid:114) κD x th (cid:19)(cid:27) + κx th (cid:21) ( Using the identity D ν +1 ( z ) − z D ν ( z ) + ν D ν − ( z ) = 0 )= − p st ( x th ) (cid:20) D D − s/k ( (cid:113) κD ) x th ) (cid:18)(cid:114) κD (cid:19) (cid:26) − (cid:18)(cid:114) κD x th (cid:19) D − s/k (cid:18)(cid:114) κD x th (cid:19) + ( − s/k ) D − s/k − (cid:18)(cid:114) κD x th (cid:19)(cid:27) + κx th (cid:21) = − p st ( x th ) (cid:20) − s (cid:114) D κ D − s/k − (cid:18)(cid:113) κD x th (cid:19) D − s/k (cid:18)(cid:113) κD x th (cid:19) (cid:21) = s (cid:114) D κ p st ( x th ) D − s/k − (cid:18)(cid:113) κD x th (cid:19) D − s/k (cid:18)(cid:113) κD x th (cid:19) (D23) a. Special threshold x th = 0 One particular case of interest is x th = 0. Substituting x th = 0 in equation (D23), we get˜ n ( x, s ) = s (cid:114) D κ p st (0) D − s/k − (0) D − s/k (0) (D24)Using the identity D ν (0) = (cid:114) π ν (cid:0) ν +12 (cid:1) (D25)the expression in equation (D24) simplifies to˜ n ( x, s ) = s (cid:114) D κ p st (0) √ + s κ )Γ (cid:0) s κ (cid:1) (D26)Using the defination of Beta function B ( a, b ) = Γ( a )Γ( b )Γ( a + b ) (D27) and indentifying a = 1 / s/ (2 κ ) and b = 1 / n ( x, s ) = s (cid:114) D κ p st (0) √ B ( s/ (2 κ ) , / /
2) (D28)The inverse Carlson-Laplace transform of the above ex-pression is straight forward and we get n ( τ ) = p st (0) (cid:114) Dκπ √ e κτ − b. Rarely visited thresholds In case of higher thresholds ( x th >> D/ κ ), the driftthat acts on the Brownian particle which makes shorterexcursion beyond the threshold x th can be approximated5as a κx ≈ κx th . So the equation (D16) simplifies to s ˜ w + ( x, s ) = κx th ∂ ˜ w + ( x, s ) ∂x + D ∂ ˜ w + ( x, s ) ∂x (D30)whose general solution is˜ w + ( x, s ) = C exp (cid:18) − xD ( κx th + (cid:112) ( κx th ) + 2 sD ) (cid:19) + C exp (cid:18) − xD ( κx th − (cid:112) ( κx th ) + 2 sD )) (cid:19) (D31)Using the boundary condition (equation D14), we get˜ w + ( x, s ) = s × exp (cid:18) ( x th − x ) D ( κx th − (cid:112) ( κx th ) + 2 sD ) (cid:19) (D32)Using it in equation (D24), we get˜ n ( s ) = p st ( x ) sDκx th + (cid:112) ( κx th ) + 2 sD (D33)The inverse Carlson-Laplace transform is given by n ( τ ) = p st ( x th )2 (cid:20) (cid:112) πτ / D e − ( κx th ) τ/ (2 D ) − ( κx th ) erfc (cid:18) κx th (cid:114) τ D (cid:19)(cid:21) (D34) Appendix E: Extremal excursions
The pdf p max ( t ; ζ | x , t ) is given by p max ( t ; ζ | x , t ) = ∂ M max ( t ; ζ | x , t ) ∂ζ = − ∂ H U ( t ; ζ | x , t ) ∂ζ Θ( ζ − x ) (E1)where we have used (84) in the second step. Therefore, (cid:104) x max ( t ) | x (cid:105) = (cid:90) ∞−∞ ζ p max ( t ; ζ | x , t ) dζ = x + (cid:90) ∞ x H U ( t ; ζ | x , t ) dζ (E2) For x = 0, from the above equation we get (cid:104) x max ( t ) | (cid:105) = (cid:90) ∞ H U ( t ; ζ | , t ) dζ . (E3)In the above equation, replacing the time variable t withthe rescaled time variable t r where t r = κt , we get (cid:104) x max ( t r ) | (cid:105) = (cid:90) ∞ H U ( t r ; ζ | , t r ) dζ (E4)Taking the the Laplace transform of both the sides (cid:104) ˜ x max ( s ) | (cid:105) = (cid:90) ∞ ˜ H U ( s ; ζ | dζ = 1 s (cid:90) ∞ F ( s/ , / , F ( s/ , / , κζ /D ) dζ (E5)where we have used the expression of ˜ H U ( s ; ζ |
0) whichwas already derived in appendix C 2 and given in equa-tion (C32). Using the following value, F ( s/ , / ,
0) = 0 (E6)and using the following expansion F ( s/ , / , κζ /D ) = 1 + s κζ D (E7)for small ζ the integral in equation (E5) simplifies to (cid:104) ˜ x max ( s ) | (cid:105) = 1 s (cid:90) ∞ dζ sκζ /D ) = π (cid:114) Dκ s − / (E8)Carrying out the inverse Laplace transform we get (cid:104) x max ( t r ) | (cid:105) = (cid:18) π Dκ t r (cid:19) / . (E9)Finally, resatoring physical time t , i.e., replacing t r by κt , we get (cid:104) x max ( t ) | (cid:105) = √ πDt . (E10) [1] Alberts, Bruce. Molecular Biology of the Cell. GarlandScience, 2008.[2] J.B.S. Haldane, On being the right size , Harper’s maga-zine (March, 1926).[3] J. T. Bonner,
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