Transport Facilitated by Rapid Binding to Elastic Tethers
TTRANSPORT FACILITATED BY RAPID BINDING TO ELASTICTETHERS ∗ BEN FOGELSON †‡ AND
JAMES P. KEENER † Abstract.
Diffusion in cell biology is important and complicated. Diffusing particles mustcontend with a complex environment as they make their way through the cell. We analyze a particulartype of complexity that arises when diffusing particles reversibly bind to elastically tethered bindingpartners. Using asymptotic analysis, we derive effective equations for the transport of both singleand multiple particles in the presence of such elastic tethers. We show that for the case of linearelasticity and simple binding kinetics, the elastic tethers have a weak hindering effect on particlemotion when only one particle is present, while, remarkably, strongly enhancing particle motionwhen multiple particles are present. We give a physical interpretation of this result that suggests asimilar effect may be present in other biological settings.
Key words. facilitated diffusion, asymptotic analysis, quasi-steady-state, elastic binding
AMS subject classifications.
1. Introduction.
Diffusion is one of the fundamental spatial processes driv-ing biological function. In many cases, it is the primary mechanism by which cellsdistribute or transport particles, both within single cells and throughout tissues.Diffusion-driven morphogen gradients, in conjunction with chemical reactions andactive transport, are responsible for directional signaling in development [2, 16]; dif-fusion of signaling molecules from the cell membrane to the nucleus is important forgene regulation [4, 10]; and within the nucleus, crowding by chromatin has a profoundeffect on the diffusion of proteins to specific DNA binding sites [5]. There is a wealthof other biological examples. A unifying feature of diffusion in cell biology is thatit occurs in a complicated environment rife with heterogeneity, complex mechanicalinteractions, and chemical binding.In this paper, we examine the motion of diffusing particles that bind to and unbindfrom other objects as they move. Classical examples of this include the buffereddiffusion of calcium ions as they reversibly bind to buffering proteins, resulting in achange to calcium’s effective diffusion coefficient[7, 18], and the facilitated diffusionof oxygen into muscle fibers, where the total inward flux of oxygen is dramaticallyamplified by the binding of oxygen and myoglobin [7, 14]. If the particle of interest andits binding partner are both free to diffuse, as is the case in both of these examples,binding can dramatically enhance particle transport. If, however, the binding partneris anchored to a fixed substrate, it is straightforward to show that in the quasi-steady-state binding limit the particle’s effective diffusion coefficient always decreases.There is an important intermediate case that, to our knowledge, has not beenstudied. This is the case where the binding partner is anchored to a surface by anelastic tether, so that the binding partner has some freedom to move but does notdiffuse over the entire domain. This situation is biologically relevant. It occurs in the ∗ Submitted to the editors September 1, 2018.
Funding:
This research has been supported in part by the National Science Foundation undergrants DMS 1515130 and DMS RTG 1148230, and in part by the Mathematical Biosciences Instituteand the National Science Foundation under grant DMS 1440386. Any opinions, findings, and conclu-sions or recommendations expressed in this material are those of the authors and do not necessarilyreflect the views of the National Science Foundation. † Department of Mathematics, University of Utah, Salt Lake City, UT 84112 ‡ Corresponding author: [email protected], http://math.utah.edu/ ∼ ben1 a r X i v : . [ q - b i o . S C ] S e p B. FOGELSON AND J. P. KEENER nuclear pore complex in the interaction between karyopherin chaperones and elasticFG nucleoporins [3, 8, 9, 13, 15], and in a similar manner in the ciliary pore complex[6, 12, 17]. It is also relevant to the diffusion of signaling molecules near membrane-bound cognate receptors, where elasticity can come from the mechanical propertiesof the receptor or from the membrane itself.The fundamental question is how the mechanical properties of the elastic tetherinfluence the particle of interest’s overall motion. In classical facilitated and buffereddiffusion, this overall motion is controlled by an interplay between the reaction ratesgoverning particle binding and the diffusion coefficients of both the particle and itsbinding partner. The elastic properties of a tethered binding partner seem likely toalter this interplay in a way that is hard to predict a priori. On the one hand, wemight intuit that the freedom of bound tethers to fluctuate about their base mightcause them to bind particles “at a distance,” and thereby exert an elastic force thataccelerates particle motion. On the other hand, a particle bound to a tether will notbe able to move very far without unbinding.In this paper, we develop and analyze a minimal model for investigating theinteractions between particle diffusion, particle-tether binding, and tether elasticity.We begin by studying the motion of a single particle diffusing in one dimension againsta background density of elastically tethered binding sites. We start with a Fokker-Planck system describing both the position and the binding state of the particle and,using a quasi-steady-state reduction, derive a reduced Fokker-Planck equation forthe effective motion of the particle in the limit of fast binding and unbinding. Ourapproach is similar to the quasi-steady-state analysis of molecular motors developedby Newby and Bressloff [11] and extended to nonlinear reaction terms by Zmurchoket. al. [19]. The key difference is that our analysis explicitly accounts for spatialvariations in the position of the binding sites, which changes the dimensionality ofour starting system of equations. Our analysis shows that the particle’s effectivemotion is controlled by a set of moments related to the space-dependent binding andunbinding rate functions. In particular, we show that for the simplest physical caseof linearly elastic tethers, the particle’s effective diffusion coefficient always drops.Next, we generalize to the case where multiple diffusing particles are present. Tostudy this, we apply a slightly modified version of a model that we recently developedfor nucleocytoplasmic transport [3]. We show that, in contrast to the single particlecase, in the limit where binding is fast and the tethers are short compared to thedomain of interest, the effective diffusion coefficient always increases. This qualitativedifference between single and multiple-particle motion is surprising, and in the finalpart of the paper we give a physical interpretation of these results.
2. One-dimensional motion of a single particle.
In this section, we derivean equation for the one-dimensional motion of a particle that undergoes diffusion whilerapidly binding to and unbinding from a continuum of elastic tethers with constantdensity, as depicted schematically in Figure 1. The starting point for this derivationis the set of differential Chapman-Kolmogorov equations ∂p u ∂t = D ∂ p u ∂x − p u (cid:90) ∞−∞ k + ( y − x ) dy + (cid:90) ∞−∞ k − ( y − x ) p b ( x, y, t ) dy, (2.1) ∂p b ∂t = D ∂ p b ∂x − ∂∂x (cid:18) kζ ( y − x ) p b (cid:19) + k + (cid:0) y − x (cid:1) p u − k − (cid:0) y − x (cid:1) p b , (2.2)where p u ( x, t ) is the probability density that the particle is unbound and at location x at time t and p b ( x, y, t ) is the probability density that the particle is at location x RANSPORT FACILITATED BY RAPID BINDING TO ELASTIC TETHERS x y Fig. 1: Schematic of single-particle diffusion mediated by elastic tethers. (a)
A singleparticle (green) diffuses slowly in one dimension. A collection of elastically tetheredbinding sites (blue) diffuses much more rapidly than the particle. (b)
Rapid diffusionof tethers means that a particle at position x has a non-zero probability per time ofbinding to a tether anchored at position y .and is bound to a tether whose base is at y . Together, (2.1) and (2.2) describe theevolution of these probabilities in terms of the particle’s diffusion and drag coefficients D and ζ , and the tether’s elastic spring constant k . The functions k + ( z ) and k − ( z )give the on and off rates for particle-tether binding, which we allow to be dependenton the spacing z between the particle and the tether base. We treat k + and k − as theoverall reaction rates after accounting for the density of tether bases, so that we do notneed to account for that density separately. The key assumption in our model is thatan unbound tether diffuses much more rapidly than the particle, so that we do notneed to track the position of individual tethers over time. Instead, we can representthe interaction between free tethers and particles with the binding rate k + ( z ), whichin general will depend on the equilibrium distribution of the tether head about itsbase. This is discussed for a linear spring in subsection 2.4.Our goal in this derivation is to describe the overall position of the particle,independent of its binding state. In other words, we seek an equation describing theevolution of the quantity(2.3) p ( x, t ) = p u ( x, t ) + (cid:90) ∞−∞ p b ( x, y, t ) dy, which is the total probability density that the particle is at position x at time t . Weseek an approximate equation for p ( x, t ) that is valid in the limit of fast binding andunbinding. To approximate the particle’s motion in thelimit of fast binding and unbinding, we nondimensionalize (2.1) and (2.2) in order
B. FOGELSON AND J. P. KEENER to identify the relevant small parameter. This requires us to choose scales ¯ L and ¯ t forlength and time. Since the relevant lengthscale may vary depending on the specificphysical application being modeled, we leave the lengthscale as ¯ L .There are three important timescales: the binding timescale(2.4) τ binding = (cid:18)(cid:90) ∞−∞ k + ( z ) dz (cid:19) − , the elastic relaxation timescale for a bound particle(2.5) τ relaxation = ζk , and the diffusive timescale(2.6) τ diffusion = ¯ L D .
We expect that in most physical situations, binding dynamics are faster than bothelastic and diffusive motion so that τ binding (cid:28) τ relaxation and τ binding (cid:28) τ diffusion . Therelationship between τ relaxation and τ diffusion is more subtle and is discussed below.We choose to nondimensionalize time in units of the timescale ¯ t = τ diffusion . Defin-ing the functions ˆ p u (ˆ x, ˆ t ) and ˆ p b (ˆ x, ˆ y, ˆ t ), where ˆ x = x/ ¯ L , ˆ y = y/ ¯ L , and ˆ z = z/ ¯ L , by p u ( x, t ) = 1¯ L ˆ p u ( x/ ¯ L, t/ ¯ t )(2.7) p b ( x, y, t ) = 1¯ L ˆ p b ( x/ ¯ L, y/ ¯ L, t/ ¯ t ) , (2.8)the system (2.1) and (2.2) becomes ∂ ˆ p u ∂ ˆ t = ∂ ˆ p u ∂ ˆ x − ˆ p u (cid:90) ∞−∞ ˆ k + (ˆ y − ˆ x ) d ˆ y + (cid:90) ∞−∞ ˆ k − (ˆ y − ˆ x ) ˆ p b (ˆ x, ˆ y, ˆ t ) d ˆ y, (2.9) ∂ ˆ p b ∂ ˆ t = ∂ ˆ p b ∂ ˆ x − λ ∂∂ ˆ x (cid:16) (ˆ y − ˆ x ) ˆ p b (cid:17) + ˆ k + (ˆ y − ˆ x ) ˆ p u − ˆ k − (ˆ y − ˆ x ) ˆ p b , (2.10)where the nondimensionalized on and off rates ˆ k + and ˆ k − areˆ k + (ˆ z ) = ¯ L ¯ tk + ( ¯ L ˆ z ) , (2.11) ˆ k − (ˆ z ) = ¯ tk − ( ¯ L ˆ z ) , (2.12)and the non-dimensional parameter λ is(2.13) λ = τ relaxation τ diffusion . The size of λ can be estimated as follows. We know from the Stokes-Einstein relationthat Dζ = k B T , so that λ can be rewritten as(2.14) λ = k B Tk ¯ L , the ratio of thermal energy to the elastic energy stored in a spring stretched deformedby ¯ L . If this quantity is large, then the tether springs must be very weak on the RANSPORT FACILITATED BY RAPID BINDING TO ELASTIC TETHERS τ relaxation <τ diffusion , and so λ < k + and ˆ k − to be large, and so we intro-duce the nondimensional parameter ε = τ binding /τ diffusion and define the rescaled ratefunctions α and β by α (ˆ z ) = ε ˆ k + (ˆ z ) , (2.15) β (ˆ z ) = ε ˆ k − (ˆ z ) . (2.16)Note that our definition of ε means that (cid:82) ∞−∞ α (ˆ z ) d ˆ z = 1, so we can write thenondimensional system of equations as ∂ ˆ p u ∂ ˆ t = ∂ ˆ p u ∂ ˆ x − ε ˆ p u + 1 ε (cid:90) ∞−∞ β (ˆ y − ˆ x ) ˆ p b (ˆ x, ˆ y, ˆ t ) d ˆ y, (2.17) ∂ ˆ p b ∂ ˆ t = ∂ ˆ p b ∂ ˆ x − λ ∂∂ ˆ x (cid:16) (ˆ y − ˆ x ) ˆ p b (cid:17) + 1 ε α (ˆ y − ˆ x ) ˆ p u − ε β (ˆ y − ˆ x ) ˆ p b . (2.18)In the following sections, we develop an approximate expression for the positionof the particle in the limit ε (cid:28) After dropping the hats over our nondimensional functions andvariables, we seek to approximate the system of equations ∂p u ∂t = ∂ p u ∂x − ε p u + 1 ε (cid:90) ∞−∞ β ( y − x ) p b ( x, y, t ) dy, (2.19) ∂p b ∂t = ∂ p b ∂x − λ ∂∂x (cid:16) ( y − x ) p b (cid:17) + α ( y − x ) ε p u − β ( y − x ) ε p b (2.20)for ε (cid:28) ε → p b = α ( y − x ) β ( y − x ) p u .Motivated by this, and using vector notation to describe the solution [ p u , p b ] T , weintroduce the reaction operator R , defined by(2.21) R (cid:18)(cid:20) p u p b (cid:21)(cid:19) = (cid:20) − p u + (cid:82) ∞−∞ β ( y − x ) p b dyα ( y − x ) p u − β ( y − x ) p b (cid:21) . This allows us to split the solution into a quasi-steady-state term [ q u , q b ] T in thenullspace of R and a remainder term [ r u , r b ] T in the range of R:(2.22) (cid:20) p u p b (cid:21) = (cid:20) q u q b (cid:21) + (cid:20) r u r b (cid:21) . The nullspace of R is spanned by [1 , α ( y − x ) /β ( y − x )] T . Defining the function κ ( z ) = α ( z ) /β ( z ), the quasi-steady-state term can be written in terms of a scalarvariable q ( x, t ) as(2.23) (cid:20) q u ( x, t ) q b ( x, y, t ) (cid:21) = 11 + K (cid:20) κ ( y − x ) (cid:21) q ( x, t ) , where K = (cid:82) ∞−∞ κ ( z ) dz , and the prefactor K is a useful normalization. B. FOGELSON AND J. P. KEENER
For [ r u , r b ] T to be in the range of R , we know from the Fredholm alternative thatit must be orthogonal to the nullspace of the adjoint operator R ∗ . It is straightforwardto show that the adjoint operator is(2.24) R ∗ (cid:18)(cid:20) p u p b (cid:21)(cid:19) = (cid:20) − p u + (cid:82) ∞−∞ α ( y − x ) p b dyβp u − βp b (cid:21) , where the appropriate inner product is(2.25) (cid:28)(cid:20) f u f b (cid:21) , (cid:20) g u g b (cid:21)(cid:29) = f u g u + (cid:90) ∞−∞ f b g b dy. The nullspace of R ∗ is spanned by the vector [1 , T , and so the remainder term mustsatisfy the orthogonality condition(2.26) r u + (cid:90) ∞−∞ r b dy = 0 . This means we can write the remainder [ r u , r b ] T in terms of a single unknown function r ( x, y, t ), with r b ( x, y, t ) = r ( x, y, t ) and r u ( x, t ) = − (cid:82) ∞−∞ r ( x, y, t ) dy .Substituting this splitting into (2.19) and (2.20), integrating (2.20) over y , andadding that to (2.19), we get the equation(2.27) ∂q∂t = ∂ q∂x − K λ (1 + K ) ∂q∂x − λ ∂∂x (cid:18)(cid:90) ∞−∞ ( y − x ) r ( x, y, t ) dy (cid:19) , where K = (cid:82) ∞−∞ zκ ( z ) dz . Using this expression for ∂q∂t in (2.20), we also get(2.28) ε ∂r∂t = − α ( y − x ) (cid:90) ∞−∞ r ( x, y, t ) dy − β ( y − x ) r ( x, y, t )+ ε (cid:34) ∂ r∂x − λ ∂∂x (cid:16) ( y − x ) r ( x, y, t ) (cid:17) + κ ( y − x ) λ (1 + K ) ∂∂x (cid:90) ∞−∞ ( y − x ) r ( x, y, t ) dy + 11 + K ∂ ∂x (cid:16) κ ( y − x ) q (cid:17) −
11 + K κ ( y − x ) ∂ q∂x + K λ (1 + K ) κ ( y − x ) ∂q∂x − λ (1 + K ) ∂∂x (cid:16) ( y − x ) κ ( y − x ) q (cid:17)(cid:35) . It is important to note that (2.27) and (2.28) are exact. There is no approximationin going from (2.19) and (2.20) to (2.27) and (2.28), only a change of variables from p u and p b to q and r . Note also that the orthogonality condition (2.26) implies that(2.29) p u ( x, t ) + (cid:90) ∞−∞ p b ( x, y, t ) dy = q ( x, t ) , so that q is the total probability density that the particle is at position x . This wasthe reason for the prefactor of K in the definition of q in (2.23). RANSPORT FACILITATED BY RAPID BINDING TO ELASTIC TETHERS We seek an approximate equation for q that is valid when ε (cid:28)
1. To do this, we expand r as a power series in ε :(2.30) r ( x, y, t ) = r ( x, y, t ) + εr ( x, y, t ) + O ( ε ) , and solve for r and r . From (2.28), the zeroth order equation is(2.31) 0 = − α ( y − x ) (cid:90) ∞−∞ r ( x, y, t ) dy − β ( y − x ) r ( x, y, t ) . Dividing (2.31) by β ( y − x ) and integrating gives(2.32) (1 + K ) (cid:90) ∞−∞ r ( x, y, t ) dy = 0 . Substituting this back into (2.31), we get r ( x, y, t ) = 0. This confirms that, asexpected, the remainder term r is small when the reactions are fast. Substituting r = 0 into (2.28) gives the first order equation (2.33) 0 = − (1 + K ) (cid:16) r + κ ( y − x ) (cid:90) ∞−∞ r ( x, y, t ) dy (cid:17) + 1 β ( y − x ) ∂ ∂x (cid:16) κ ( y − x ) q (cid:17) − κ ( y − x ) β ( y − x ) ∂ q∂x − λβ ( y − x ) ∂∂x (cid:16) ( y − x ) κ ( y − x ) q (cid:17) + K λ (1 + K ) κ ( y − x ) β ( y − x ) ∂q∂x . We can integrate this equation over y to solve for (cid:82) ∞−∞ r . Noting that all the integralsshare a similar form, we write(2.34) φ i,j,k ( z ) = z i β ( z ) ∂ j ∂z j (cid:16) z k κ ( z ) (cid:17) for integers i , j , and k , and let Φ i,j,k = (cid:82) ∞−∞ φ i,j,k ( z ) dz .In terms of Φ, the integral of r is(2.35) (cid:90) ∞−∞ r ( x, y, t ) dy = 1 λ (1 + K ) (cid:20)(cid:16) λ Φ , , + Φ , , (cid:17) q ( x, t ) − (cid:16) λ Φ , , + Φ , , − K K Φ , , (cid:17) ∂q∂x (cid:21) . Substituting this back into (2.33) lets us solve for r . Finally, substituting for r in(2.27), we get a single differential equation for q : (2.36) ∂q∂t = ∂ q∂x − K λ (1 + K ) ∂q∂x + ελ (1 + K ) (cid:34)(cid:18) K K (cid:0) Φ , , + λ Φ , , (cid:1) − (cid:0) Φ , , + λ Φ , , (cid:1)(cid:19) ∂q∂x + (cid:18) Φ , , + 2 λ Φ , , − K K (cid:16) λ Φ , , + Φ , , (cid:17) + (cid:18) K K (cid:19) Φ , , (cid:19) ∂ q∂x (cid:35) + O ( ε ) . Equation (2.36) is our desired result. This equation describes the evolution of theparticle position to order ε purely in terms of q and integrals of the binding on and offrate functions. In the following sections, we analyze (2.36) in some important specialcases. B. FOGELSON AND J. P. KEENER
Equation (2.36) simplifies sub-stantially when the binding and unbinding rates are spatially symmetric. When thisis true, both α ( z ) and β ( z ) are even functions of z . This causes many of the integralterms in (2.36) to evaluate to zero. Specifically,(2.37) K = (cid:90) ∞−∞ zκ ( z ) dz = (cid:90) ∞−∞ z α ( z ) β ( z ) dz = 0 , and(2.38) Φ i,j,k = (cid:90) ∞−∞ z i β ( z ) ∂ j ∂z j (cid:16) z k κ ( z ) (cid:17) dz = 0for i + j + k odd. Then, to order ε , the equation for q becomes(2.39) ∂q∂t = ∂ q∂x + ελ (1 + K ) (cid:16) Φ , , + 2 λ Φ , , (cid:17) ∂ q∂x . Thus, when binding and unbinding are spatially symmetric, the net motion of theparticle is purely diffusive. Moreover, fast binding to and unbinding from elastic teth-ers is capable of increasing or decreasing the particle’s effective diffusion coefficient,depending on the sign of Φ , , +2 λ Φ , , , which in turn depends on the specific choicesof the reaction rate functions α and β as well as the ratio λ = τ relaxation /τ diffusion . The simplest binding ratesoccur when the elastic tether is a simple linear spring and when the diffusion coefficientof a free tether is much larger than that of the particle. In this case, the dimensionalsteady-state probability p tether ( x ; y ) that the tether’s free end is at position x withbase fixed at y satisfies the differential equation(2.40) 0 = D tether ∂ p tether ∂x − k tether ζ tether ∂∂x (cid:16) ( y − x ) p tether (cid:17) , and so p tether is Gaussian:(2.41) p tether ( x ; y ) = 1 √ π (cid:115) k tether D tether ζ tether e − k tether D tether ζ tether ( y − x ) . Here D tether , ζ tether , and k tether are the diffusion, drag, and spring coefficients of thefree tether. The Stokes-Einstein relation tells us that diffusion and drag are relatedby D tether = k B Tζ tether , so(2.42) p tether ( x ; y ) = 1 √ π (cid:114) k tether k B T e − k tether kBT ( y − x ) . We return to p tether in a moment, but first we need to consider the basic kineticsof particle-tether binding. If the reaction simply occurred in a well-mixed solutionof particles and tether binding sites (without the elastic anchors), the system wouldevolve according to mass-action kinetics as(2.43) dc complex dt = k on c particle c tether − k off c complex , for linear densities of particles, tether binding sites, and particle-tether complexes c particle , c tether , and c complex , and rate constants k on and k off . We can use the rate RANSPORT FACILITATED BY RAPID BINDING TO ELASTIC TETHERS k on along with p tether to compute the separation-dependent binding ratefunction k + .Let the linear density of tether bases anchored to the domain be ρ . For a smalllength dy the quantity ρ p tether ( x ; y ) dy can be interpreted as the expected density oftether free ends at x with bases in the interval ( y, y + dy ). Since p u is the probabilitydensity that there is a free particle at x , the quantity k on ρ p tether ( x ; y ) dy p u ( x, t ) isthe rate of increase of the probability density that there is a particle-tether complexwith base in ( y, y + dy ) at position x . By definition, that is the rate of increase of thequantity p b ( x, y, t ) dy . Recalling (2.2), our original Fokker-Planck equation for theevolution of p b , the kinetic rate constant k on is thus related to the rate function k + according to(2.44) k + ( y − x ) = k on ρ p tether ( x ; y ) = k on ρ √ π (cid:114) k tether k B T e − k tether kBT ( y − x ) . In nondimensional variables, this gives us the binding rate function(2.45) α ( y − x ) = (cid:114) k tether πkλ e − k tether2 kλ ( y − x ) . Note that ρ and k + do not appear in (2.45). This is because α was defined by α ( · ) = ε ˆ k + ( · ) = τ binding τ diffusion ˆ k + ( · ), and τ binding = (cid:16)(cid:82) ∞−∞ k + ( z ) dz (cid:17) − = 1 / ( ρk on ). The quantity k tether /k is the ratio of the free tether spring coefficient to the bound tether springcoefficient. These two spring coefficients could be different due to conformationalchanges that occur upon binding. Taking the unbind-ing rate to be constant, we have the dimensional expression(2.46) k − ( z ) = k off , which is equivalent to the nondimensional expression(2.47) β ( z ) = β , where β = τ binding k off = k off ρk on .The evolution equation (2.39) for q simplifies to(2.48) ∂q∂t = ∂ q∂x + ε ν − β (1 + β ) λ ∂ q∂x , where ν = k/k tether is the ratio of bound to unbound tether spring coefficients. Equa-tion (2.48) predicts that for elastic tethers to increase the diffusion coefficient of asingle particle, the inequality k > k tether must be satisfied, meaning that the tethermust be at least twice as stiff when bound to a particle than when unbound. It is easyto envision how, in a biological setting, this could be the case: binding could induce aconformational change that stiffens the tether. In such a case, we expect that eitherthe binding or unbinding step would require an input of chemical energy in the formof ATP. If, on the other hand, there is no conformational change due to particle-tetherbinding (so k = k tether and therefore ν = 1), (2.48) predicts that binding will hinderthe particle’s effective motion.0 B. FOGELSON AND J. P. KEENER
Another reasonable choice for k − ( z ) would be a classic slip bond with force-dependent unbinding [1]:(2.49) k − ( z ) = k off e (cid:0) k | z | /F unbinding (cid:1) , for some characteristic unbinding force F unbinding . In nondimensional units, (2.49)becomes(2.50) β ( z ) = β e (cid:0) | z | /γ (cid:1) , where β = k off τ binding and γ = F unbinding / ( k ¯ L ). With this unbinding rate function,the evolution equation for q becomes(2.51) ∂q∂t = ∂ q∂x + (cid:15) √ − ν ) ν (cid:113) λνγ + √ πνe λνγ (cid:16) λ ( ν − νγ + ν − (cid:17) erfc (cid:16)(cid:113) λνγ (cid:17) √ πβ λν (cid:16) β + e λν γ erfc (cid:16)(cid:113) λν γ (cid:17)(cid:17) ∂ q∂x . While this equation is clearly more complicated than when β ( z ) = β , the qualitativepredictions are similar. Note that the numerator of the correction term in (2.51) de-pends only on ν and the quantity γ /λ . Using this, we plot the parameter regimes forwhich (2.51) predicts enhanced and hindered diffusion in Figure 2. We can interpret γ /λ by rewriting it as (cid:0) F /k (cid:1)(cid:14) k B T . The numerator here is the work it takesto stretch the tether by length F unbinding /k , which is the characteristic lengthscaleat which the bond begins to slip. Thus, as Figure 2 shows, the larger the unbindinglengthscale the easier it is for diffusion to be enhanced. Indeed, in the limit as γ → ∞ ,(2.51) reduces to (2.48), for which diffusion is enhanced when ν >
2. Additionally,when ν = 1, diffusion is hindered regardless of the value of γ /λ . This is easily verifiedalgebraically by substituting ν = 1 into (2.51).The main conclusions of this section are first, that for general spatially-dependentbinding and unbinding rates, it is possible for a particle’s overall motion to be en-hanced by binding to a population of elastic tethers; and second, that in the simplestcase where the tether mechanical properties are fixed and binding is governed by thelinear elasticity of the tethers, the particle’s diffusion is always reduced. A naturalquestion, which we address in the following section, is whether there is some way toenhance particle motion even in the simple case.
3. Motion of multiple particles.
In [3], we proposed a model for enhancedtransport through the nuclear pore that relied on competition between two distinctspecies of particles for binding to a population of elastic tethers. Enhanced nucleartransport in that case depended on a set of biochemically motivated boundary con-ditions as well as the interaction between two distinct species of moving particles.Here, we derive a version of that model but with only one chemical species andwith a scaling consistent with our analysis in section 2. We show that even whenthe elastic tethers are simple linear springs, intraspecies competition is sufficient tosubstantially enhance particle diffusion. Even more remarkably, we show that thisenhancement effect is O (1). The derivation in this section follows the general ap-proach we developed in [3]. We proceed in three main steps: first, for a given spatialdistribution v ( x, t ) of particles we derive an expression for the probability density thata tether anchored at y is bound to a particle at x . Second, for an ensemble of tethers RANSPORT FACILITATED BY RAPID BINDING TO ELASTIC TETHERS Enhanced di ff usionHindered di ff usion0 5 10 15 2005101520 Fig. 2: Parameter regimes for enhanced and hindered diffusion with slip bond un-binding given by (2.50).we estimate the expected force exerted on each particle. Third, we approximate thisforce to obtain a closed form expression for the flux of particles. From this flux wecan write down a conservation law for the evolution of the particle distribution.
Let v ( x, t ) be theconcentration (number per unit length) of particles at position x and time t , andconsider a single elastic tether anchored at position y . We can write an equationdescribing the probability of finding the tether bound to a particular particle:(3.1) ∂p v ∂t = 1 ρ k + (cid:0) y − x (cid:1) v ( x, t ) p f ( t ; y ) − k − (cid:0) y − x ) (cid:1) p v ( x, t ; y ) . Here, p v ( x, t ; y ) is the probability density that the tether anchored at y is bound toa particle at x at time t , and p f ( t ; y ) is the probability that the tether at position y is free. We have written the tether base coordinate y as a parameter rather thanan independent variable to emphasize that we are considering a single tether fixed at y . The binding rate includes a factor of ρ because our definition of k + in section 2incorporates the effect of tether density, while in (3.1) we are considering a singletether.Note also that since our probabilities must sum to 1, we have(3.2) p f ( t ; y ) = 1 − (cid:90) ∞−∞ p v ( x, t ; y ) dx. B. FOGELSON AND J. P. KEENER
We can non-dimensionalize (3.1) and (3.2) with the same scaling as in section 2: (cid:15) ∂ ˆ p v ∂ ˆ t = ¯ vρ α (ˆ y − ˆ x )ˆ v (ˆ x, ˆ t ) ˆ p f (ˆ t ; ˆ y ) − β (ˆ y − ˆ x ) ˆ p v (ˆ x, ˆ t ; ˆ y ) , (3.3) ˆ p f (ˆ t ; ˆ y ) = 1 − (cid:90) ∞−∞ ˆ p v (ˆ x, ˆ t ; ˆ y ) d ˆ x. (3.4)where ¯ v is a concentration scale for the particles. Since (cid:15) (cid:28)
1, we take p v to be inquasi-steady-state:(3.5) ˆ p v (ˆ x ; ˆ y ) = κ (ˆ y − ˆ x )ˆ v (ˆ x, ˆ t ) ρ ¯ v + (cid:82) ∞−∞ κ (ˆ y − ˆ x )ˆ v (ˆ x, ˆ t ) d ˆ x . Equation (3.5) can be interpreted as the fraction of time that an isolated tetheranchored at ˆ y spends bound to a particle at position ˆ x . When more than one tetheris present, we expect there to be correlations between the binding states of differenttethers. We ignore those correlations, and treat (3.5) as a mean-field expression forthe fraction of time that a tether at ˆ y spends bound to a particle at ˆ x even whenother tethers are present. This approximation lets us estimate the average forceon each particle due to tether binding. For a constant linear density of tethers ρ , thetotal dimensional force on all particles at position x is(3.6) F ( x, t ) = ρ (cid:90) ∞−∞ k ( y − x ) p v ( x ; y ) dy. For problems on this physical scale, the natural measure of energy is k B T , and sothe natural scale for force is ¯ F = k B T / ¯ L . With this scaling, the nondimensionalexpression for the force is(3.7) ˆ F (ˆ x, ˆ t ) = ¯ Lρλ (cid:90) ∞−∞ (ˆ y − ˆ x ) κ (ˆ y − ˆ x )ˆ v (ˆ x, ˆ t ) ρ ¯ v + (cid:82) ∞−∞ κ (ˆ z )ˆ v (ˆ z, ˆ t ) d ˆ z d ˆ y. With the expressions for α ( · ) and β ( · ) from (2.45) and (2.47), this becomes(3.8) ˆ F (ˆ x, ˆ t ) = ¯ Lρλ (cid:90) ∞−∞ β √ πλ (ˆ y − ˆ x ) exp (cid:0) − (ˆ y − ˆ x ) / λ (cid:1) ˆ v (ˆ x, ˆ t ) ρ ¯ v + β √ πλ (cid:82) ∞−∞ exp (cid:0) − (ˆ y − ˆ z ) / λ (cid:1) ˆ v (ˆ z, ˆ t ) d ˆ z d ˆ y. The width of the Gaussians in (3.8) is governed by the parameter λ , which we recall is τ relaxation /τ diffusion , the ratio of the elastic relaxation time to the time it takes for a freeparticle to have diffused a mean-squared distance of ¯ L . We have already argued thatthe physically interesting case is λ <
1. Now we assume that λ (cid:28)
1, and approximatethe integrals in (3.8) for small λ .Letting ˆ Z = (ˆ y − ˆ x ) / √ λ , we can rewrite the inner integral in (3.8):(3.9) (cid:90) ∞−∞ exp (cid:0) − (ˆ y − ˆ z ) / λ (cid:1) ˆ v (ˆ z, ˆ t ) d ˆ z = (cid:90) ∞−∞ √ λ exp (cid:0) − ˆ Z / (cid:1) ˆ v (ˆ y + √ λ ˆ Z, ˆ t ) d ˆ Z ≈ √ πλ ˆ v (ˆ y, ˆ t ) + O (cid:0) λ / (cid:1) . Using a similar expansion for the outer integral, we get the expected force(3.10) ˆ F (ˆ x, ˆ t ) ≈ − ¯ Lρ ˆ v (ˆ x, ˆ t )ˆ v ˆ x (ˆ x, ˆ t ) (cid:0) β ρ ¯ v + ˆ v (ˆ x, ˆ t ) (cid:1) + O ( λ ) . RANSPORT FACILITATED BY RAPID BINDING TO ELASTIC TETHERS vvv . The quantity F ( x ) is the expected total force on all particles atposition x . This force will cause the particles to move. Recalling that ζ is the particledrag coefficient, the expected flux of particles due to this elastic force is ζ F ( x ). Thetotal flux of particles is a combination of this elastic flux and normal Fickian diffusion:(3.11) J ( x, t ) = − Dv x ( x, t ) + 1 ζ F ( x, t ) . This leads to the conservation law(3.12) ∂v∂t = D ∂∂x (cid:18) − v x + F ( x, t ) k B T (cid:19) . In non-dimensional variables, this becomes(3.13) ∂ ˆ v∂ ˆ t = ∂∂ ˆ x (cid:18) − ˆ v ˆ x + ˆ F (ˆ x, ˆ t )¯ L ¯ v (cid:19) , and using (3.10) for the force ˆ F , the conservation law becomes(3.14) ∂ ˆ v∂ ˆ t = − ∂∂ ˆ x ˆ v ˆ x + ρ ¯ v ˆ v ˆ v ˆ x (cid:16) β ρ ¯ v + ˆ v (cid:17) Choosing the concentration scale ¯ v = k off k on , and recalling that β = k off ρk on , we arrive atthe equation(3.15) ∂ ˆ v∂ ˆ t = − ∂∂ ˆ x (cid:18) ˆ v ˆ x + 1 β ˆ v ˆ v ˆ x (1 + ˆ v ) (cid:19) . Equation (3.15) can be interpreted as a nonlinear dif-fusion equation with the concentration-dependent diffusion coefficient 1+ β ˆ v/ (1+ˆ v ) .This predicts that when (cid:15) (cid:28) λ (cid:28)
1, diffusion is always enhanced by the compet-itive interaction between multiple particles and a density of elastic tethers. Interest-ingly, the form of the diffusion coefficient suggests an optimal particle concentrationfor enhanced diffusion. The diffusion coefficient grows like 1 + ˆ v/β for ˆ v (cid:28)
1, reachesa maximum of 1 + β when ˆ v = 1, and decays back to 1 as ˆ v → ∞ .This prediction of enhanced diffusion is in sharp contrast to the hindered diffusionin the single-particle case, where we found a diffusion coefficient of 1 − ε β (1+ β ) .This contrast is striking for two reasons. First, there is the simple fact that the samecollection of elastic tethers slows down the motion of a single particle while speedingup the motion of an ensemble. Second, the magnitudes of the two effects are different:the hindrance in the single-particle case is only O ( ε ), while the enhancement in themultiple-particle case is O (1). At first glance, it seems counter-intuitive thatthe effect of tether binding in the single and multiple particle cases would be differentin both sign and order of magnitude. We can understand the physics driving thisphenomenon by careful considering the force driving particle motion in each case.4
B. FOGELSON AND J. P. KEENER
Particles move due to a combination of Brownian motion driven by thermal fluc-tuations and directed motion driven by elastic tethers. When only one particle ispresent, as shown in Figure 3a, symmetry means that the net elastic force on the par-ticle is zero. Moreover, the particle spends most of its time bound to the tethers thatare directly under it. Since the magnitude of the elastic tether force is proportionalto the distance between the particle and the tether base, the particle feels little tono force from these tethers. Instead, these tethers anchor the particle and prevent itfrom drifting due to Brownian motion. While the particle does spend some amountof time bound to tethers that are anchored far away from the particle, the anchoringeffect of binding to nearby tethers is dominant.When multiple particles are present, competition for tether binding breaks thesymmetry of the system. Figure 3b shows the case where two particles are near oneanother. The two particles can each bind to the tethers in between them, and so thosetethers spend part of their time bound to each particle. This means that the particleon the left spends more overall time bound to tethers that are anchored to its left,and the particle on the right similarly spends more time bound to tethers to its right.Thus, the expected force on the left particle points to the left, and the average forceon the right particle points to the right. For the two particles shown in Figure 3b, thisleads to an effective repulsive force between particles. For a collection of particles,that repulsion will push particles from regions of high concentration to regions of lowconcentration.
4. Conclusion.
Understanding how particles move in the crowded, complex,and biochemically active environment of the cell is one of the major challenges ofmodern biology. We have developed a minimal model to explore one aspect of thischallenge: how diffusion is influenced by frequent and reversible binding to elasticbinding sites. Remarkably, our model predicts that the behavior of a single diffusingparticle is dramatically different than the behavior of an ensemble. While the singleparticle’s motion is hindered by binding, chemical competition within the ensemblegenerates an effective repulsion that drives particles down concentration gradients.The key insight that explains this phenomenon is the observation that whenbinding sites are attached to flexible tethers, competition for binding occurs not justamong particles at a single spatial location, but between particles at nearby locations.This nonlocal competition provides a mechanism by which particles “sense” the localconcentration gradient: they are more likely to bind to an adjacent tether on the sidewith fewer neighbors, and so they are more likely to experience a force pulling themto that side.While our model and analysis are limited to the one-dimensional case with linearsprings for tethers, the underlying mechanism is quite general. Whenever elasticfluctuations introduce the possibility of binding at a distance, the system becomessensitive to the local concentration gradient. This remains true even when the bindingdistance is quite short relative to lengthscales of interest, as was the case in ouranalysis.In deriving our multiple particle model, we chose to ignore correlations betweenthe binding states of individual tethers. It is unclear how significant these correlationsare, and so it is unknown whether including them would change the qualitative pre-dictions of our model. Future study and new analytical or computational techniquesare necessary to answer this question.To our knowledge, this quasi-steady-state analysis of a non-local representation ofbinding is new, and opens exciting avenues for future research. Equation (2.36), the
RANSPORT FACILITATED BY RAPID BINDING TO ELASTIC TETHERS (a)
A single particle spends equal time bound to particles on either side of it, so itexperiences a net elastic force. It is most likely to bind to tethers that are anchoreddirectly underneath it (orange curve), which will tend to prevent the particle fromdrifting. (b)
When two particles compete to bind tethers, they spend a smallerfraction of time bound to the tethers in between them than to other tethers (orangecurve). The net elastic force on each particle is non-zero, and causes the particles torepel one another (large arrows).reduced Fokker-Planck equation for single-particle motion, is valid for very generalbinding and unbinding rate functions, including the spatially anisotropic functionsthat might arise from complicated macromolecules such as molecular motors. Webelieve that our model framework, in both the single and multiple particle cases,will be a useful starting point for answering questions about transport in these morecomplicated settings.
References. [1]
G. I. Bell , Models for the specific adhesion of cells to cells. , Science, 200 (1978),pp. 618–627, https://doi.org/10.1126/science.347575.[2]
H. A. Brooks and P. C. Bressloff , Turing mechanism for homeostatic con-trol of synaptic density during C. elegans growth , Physical Review E, 96 (2017),p. 012413, https://doi.org/10.1103/PhysRevE.96.012413.[3]
B. Fogelson and J. P. Keener , Enhanced Nucleocytoplasmic Transport due to B. FOGELSON AND J. P. KEENER
Competition for Elastic Binding Sites , Biophysical Journal, 115 (2018), pp. 108–116, https://doi.org/10.1016/j.bpj.2018.05.034.[4]
S. A. Isaacson, A. J. Mauro, and J. Newby , Uniform asymptotic approx-imation of diffusion to a small target: Generalized reaction models , PhysicalReview E, 94 (2016), https://doi.org/10.1103/PhysRevE.94.042414.[5]
S. A. Isaacson, D. M. McQueen, and C. S. Peskin , The influence of volumeexclusion by chromatin on the time required to find specific DNA binding sites bydiffusion , Proceedings of the National Academy of Sciences, 108 (2011), pp. 3815–3820, https://doi.org/10.1073/pnas.1018821108.[6]
H. L. Kee and K. J. Verhey , Molecular connections between nuclearand ciliary import processes. , Cilia, 2 (2013), p. 11, https://doi.org/10.1186/2046-2530-2-11.[7]
J. P. Keener and J. Sneyd , Mathematical Physiology , vol. 8/1 of Inter-disciplinary Applied Mathematics, Springer New York, New York, NY, 2008,https://doi.org/10.1007/978-0-387-79388-7.[8]
M. Kimura, Y. Morinaka, K. Imai, S. Kose, P. Horton, andN. Imamoto , Extensive cargo identification reveals distinct biological roles ofthe 12 importin pathways , eLife, 6 (2017), p. e21184, https://doi.org/10.7554/eLife.21184.[9]
R. Y. Lim, B. Fahrenkrog, J. Koser, K. Schwarz-Herion, J. Deng, andU. Aebi , Nanomechanical basis of selective gating by the nuclear pore complex ,Science, 318 (2007), pp. 640–643, https://doi.org/10.1126/science.1145980.[10]
A. E. Lindsay, A. J. Bernoff, and M. J. Ward , First Passage Statisticsfor the Capture of a Brownian Particle by a Structured Spherical Target withMultiple Surface Traps , Multiscale Modeling & Simulation, 15 (2017), pp. 74–109, https://doi.org/10.1137/16M1077659.[11]
J. M. Newby and P. C. Bressloff , Quasi-steady State Reduction of MolecularMotor-Based Models of Directed Intermittent Search , Bulletin of MathematicalBiology, 72 (2010), pp. 1840–1866, https://doi.org/10.1007/s11538-010-9513-8.[12]
L. B. Pedersen, J. B. Mogensen, and S. T. Christensen , Endocytic Con-trol of Cellular Signaling at the Primary Cilium , Trends in Biochemical Sciences,41 (2016), pp. 784–797, https://doi.org/10.1016/j.tibs.2016.06.002.[13]
B. Raveh, J. M. Karp, S. Sparks, K. Dutta, M. P. Rout, A. Sali, andD. Cowburn , Slide-and-exchange mechanism for rapid and selective transportthrough the nuclear pore complex , Proceedings of the National Academy of Sci-ences, 113 (2016), pp. E2489–E2497, https://doi.org/10.1073/pnas.1522663113.[14]
S. Rubinow and M. Dembo , The facilitated diffusion of oxygen by hemoglobinand myoglobin , Biophysical Journal, 18 (1977), pp. 29–42, https://doi.org/10.1016/S0006-3495(77)85594-X.[15]
H. B. Schmidt and D. G¨orlich , Transport Selectivity of Nuclear Pores, PhaseSeparation, and Membraneless Organelles , Trends in Biochemical Sciences, 41(2016), pp. 46–61, https://doi.org/10.1016/j.tibs.2015.11.001.[16]
S. Y. Shvartsman and R. E. Baker , Mathematical models of morphogengradients and their effects on gene expression , Wiley Interdisciplinary Reviews:Developmental Biology, 1 (2012), pp. 715–730, https://doi.org/10.1002/wdev.55.[17]
D. Takao and K. J. Verhey , Gated entry into the ciliary compartment , Cel-lular and Molecular Life Sciences, 73 (2016), pp. 119–127, https://doi.org/10.1007/s00018-015-2058-0.[18]
J. Wagner and J. Keizer , Effects of rapid buffers on Ca2+ diffusion andCa2+ oscillations , Biophysical Journal, 67 (1994), pp. 447–456, https://doi.org/
RANSPORT FACILITATED BY RAPID BINDING TO ELASTIC TETHERS