Main

Various motor proteins such as kinesins or myosins are able to convert the chemical fuel provided by ATP into mechanical work by interacting with the semiflexible oriented filaments (mainly F-actin and microtubules) of the cytoskeleton1. As many molecules or larger cellular organelles such as vesicles, lysosomes or mitochondria, hereafter referred to as tracer particles, can randomly bind and unbind to motors, the overall transport of a tracer in the cell can be described as alternating phases of standard diffusive transport and phases of active directed transport powered by motor proteins1,15,16. Active transport in cells has been extensively studied both experimentally, for instance by single-particle tracking methods11,12, and theoretically by evaluating the mean displacement of a tracer13,17, or stationary concentration profiles14.

On the other hand, most cell functions are regulated by coordinated chemical reactions that involve low concentrations of reactants (such as ribosomes or vesicles carrying targeted proteins), and are therefore limited by transport. However, up to now a general quantitative analysis of the impact of active transport on reaction kinetics in cells, and more generally in generic active media, is still missing, even though a few specific examples have been tackled18. Here, we propose an analytical model that enables us to determine for the first time the kinetic constant of transport-limited reactions in active media.

The model relies on the idea of intermittent dynamics introduced in the context of search processes19,20,21,22,23,24,25,26,27. We consider a tracer particle evolving in a d-dimensional space (in practice d=1,2,3) that exhibits thermal diffusion phases of diffusion coefficient D (denoted phases 1), randomly interrupted by ballistic excursions bound to motors (referred to as phases 2) of constant velocity v and direction pointing in the solid angle ωv (Fig. 1a). The distribution of the filaments’ orientation is denoted by ρ(ωv), and will be taken as either disordered or polarized (Fig. 1a), which schematically reproduces the different states of the cytoskeleton1. The random duration of each phase i is assumed to be exponentially distributed with mean τi. The tracer T can react with reactants R (supposed immobile) only during free-diffusion phases 1, as T is assumed to be inactive when bound to motors, which is realized for instance when the reactants are membrane proteins (Fig. 1b,c). Reaction occurs with a finite probability per unit of time k when the tracer–reactant distance is smaller than a given reaction radius a. In what follows, we explicitly determine the kinetic constant K of the reaction T+R→R.

Figure 1: Model of reaction kinetics in active media, and examples of low-dimensional structures in biological cells.
figure 1

a, The reactant alternates thermal diffusion phases (regime 1 in red) of mean duration τ1 and diffusion coefficient D, and ballistic phases of velocity v powered by molecular motors (regime 2 in blue) of mean duration τ2. Here, the cytoskeletal filaments (in black) are in a disordered state. The polarized nematic state would correspond to parallel filaments, and is equivalent in a first approximation to a one-dimensional situation (see b) with the same concentration c1d=c3d=a3d3/b3d3 and an effective reaction radius a1d=a3dc3d2/3. Molecular motors are not represented. b, Tubular structures in cells such as axons and dendrites (d=1). c, Planar structures such as membranes and lamellipodia (d=2).

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We now present the basic equations in the case of a reactant centred in a spherical domain of radius b with a reflecting boundary. This geometry both mimics the relevant situation of a single target and provides a mean-field approximation of the general case of randomly located reactants with concentration c=ad/bd, where b is the typical distance between reactants. We start from a mean-field approximation of the first-order reaction constant28 and write K=1/〈t〉, where the key quantity of our approach is the reaction time 〈t〉, which is defined as the mean first-passage time29,30 (MFPT) of the tracer at a reactant position uniformly averaged over its initial position. For the active intermittent dynamics defined above, the MFPT of the tracer at a reactant position satisfies the following backward equation29:

where t1 is the MFPT starting in phase 1 at position r, and t2 is the MFPT starting in phase 2 at position r with velocity v. Ia is the indicator function of the ball of radius a. As these equations (1) are of integro-differential type, standard methods of resolution are not available for a general distribution ρ.

However, in the case of a disordered distribution of filaments (ρ(ωv)=1/Ωd, where Ωd is the solid angle of the d-dimensional sphere), we can use a generalized version of the decoupling approximation introduced in ref. 23 to obtain a very good approximate solution of equations (1), as described in the Methods section. Here, we present simplified expressions of the resulting kinetic constant by taking alternatively the limit , which corresponds to the ideal case of perfect reaction, and the limit D→0, which enables us to isolate the k dependence.

First we discuss the d=3 disordered case (Fig. 1a), which provides a general description of the actin cytoskeleton of a cell in nonpolarized conditions, or of a generic in vitro active solution. An analytical form of the reaction rate K3d is given in the Methods section, and is plotted in Fig. 2a,b. Strikingly, K3d can be maximized (Fig. 2a,b) as soon as the reaction radius exceeds a threshold ac≃D/v for the following value of the mean interaction time with motors:

where x0 is the solution of 2tanh(x)−2x+xtanh(x)2=0. The τ1 dependence is very weak, but we can roughly estimate the optimal value by τ1,3dopt≃6D/v2. This in turn gives the maximal reaction rate

so that the gain with respect to the reaction rate K3dp in a passive medium is G3d=K3dm/K3dp≃C a v/D with C≃0.26.

Figure 2: Optimization of the reaction rate.
figure 2

a,b, Gain of reactivity due to active transport G3d for d=3 as a function of τ2 for different values of the ratio b/a (logarithmic scale). The analytical form obtained in the Methods section (lines) is plotted against numerical simulations (symbols) for the following values of the parameters (arbitrary units): a=1.5 (brown), a=4.5 (red), a=7.5 (blue), a=10.5 (green), a=15 (yellow), with τ1=6, v=1 and D=1. K3d presents a maximum only for a>ac≃4. Standard cellular conditions (as discussed in the text) correspond to green and yellow curves for b/a=40. c,d, Gain of reactivity due to active transport G1d for d=1 as a function of τ2 (c) and τ1 (d) (logarithmic scale). The analytical form obtained in the Methods section (lines) is plotted against the exact solution (symbols), for the following values of the parameters (arbitrary units): D=1 and v=1 for all curves and a=10,b=104 (red), a=10,b=103 (blue) and a=2.5,b=103 (green). Standard cellular conditions (as discussed in the text) correspond to blue and red curves.

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Several comments are in order. (1) τ2,3dopt neither depends on D, nor on the reactant concentration. A similar analysis for k finite (in the D→0 limit) shows that this optimal value does not depend on k either, which proves that the optimal mean interaction time with motors is widely independent of the parameters characterizing the diffusion phase 1. (2) The value ac should be discussed. In standard cellular conditions, D ranges from ≃10−2 μm2 s−1 for vesicles to ≃10 μm2 s−1 for small proteins, whereas the typical velocity of a motor protein is v≃1 μm s−1, a value that is widely independent of the size of the cargo1. This gives a critical reaction radius ac ranging from ≃10 nm for vesicles, which is smaller than any cellular organelle, to ≃10 μm for single molecules, which is comparable to the whole cell dimension. Hence, this shows that in such a three-dimensional disordered case, active transport can optimize reactivity for sufficiently large tracers such as vesicles, as motor-mediated motion permits a fast relocation to unexplored regions, whereas it is inefficient for standard molecular reaction kinetics, mainly because at the cell scale molecular free diffusion is faster than motor-mediated motion. This could help justify that many molecular species in cells are transported in vesicles. Interestingly, in standard cellular conditions τ2,3dopt is of the order of 0.1 s for a typical reaction radius of the order of 0.1 μm. This value is compatible with experimental observations1, and suggests that cellular transport is close to optimum. (3) The typical gain for a vesicle of reaction radius a≳0.1 μm in standard cellular conditions is G3d≳2.5 (Fig. 2a,b) and can reach G3d≳10 for faster types of molecular motor such as myosins (v≃4 μm s−1, see refs 1, 11), independently of the reactant concentration c. As we shall show below, the gain will be significantly higher in lower-dimensional structures such as axons.

We now come to the d=2 disordered case (Fig. 1c). Striking examples in cells are given by the cytoplasmic membrane, which is closely coupled to the network of cortical actin filaments, or the lamellipodium of adhering cells1. In many cases, the orientation of filaments can be assumed to be random. This problem then exactly maps the search problem studied in ref. 23, where the reaction time was calculated. This enables us to show that as for d=3, the reaction rate K2d can be optimized in the regime D/v≪a≪b. Remarkably, the optimal interaction time τ2,2dopt takes the same value in the two limits and D→0 :

which indicates that again τ2,2dopt does not depend on the parameters of the thermal diffusion phase, neither through D nor k. In the limit , we have τ1,2dopt=(D/2v2)(ln2(1/c1/2)/(2ln(1/c1/2)−1)), and the maximal reaction rate can then be obtained:

Comparing this expression to the case of passive transport yields a gain . As in the d=3 case, this proves that active transport enhances reactivity for large enough tracers (with a critical reaction radius ac≃D/v of the same order as in the d=3 case) such as vesicles. However, here the gain G2d depends on the reactant concentration c, and can be more significant: with the same values of D, v and a as given above for a vesicle in standard cellular conditions, and for low concentrations of reactants (such as specific membrane receptors) with a typical distance between reactants b≳10 μm, the typical gain is G2d≳8, and reaches 10 for single reactants (such as some signalling molecules).

The case of nematic order of the cytoskeletal filaments, which depicts for instance the situation of a polarized cell1, can be shown to be equivalent in a first approximation to the one-dimensional case, which is exactly solvable (Fig. 1a,b). The d=1 case is also important on its own in cell biology, as many one-dimensional active structures such as axons, dendrites or stress fibres1 are present in living cells. As an illustration, we take the example of an axon, filled with parallel microtubules pointing their plus end in a direction e. We consider a tracer particle interacting with both kinesins (‘+’ end directed motors, of average velocity v e) and dyneins (‘−’ end directed motors, of average velocity −v e) with the same characteristic interaction time τ2 (see Fig. 1b). For this type of tracer, the MFPT satisfies equations (1) with an effective nematic distribution of filaments ρ(ωv)=(1/2)(δ(v−e)+δ(v+e)). The reaction rate K1d is obtained exactly in this case (see the Methods section), and is maximized in the regime D/v≪a≪b for the following values of the characteristic times (see Fig. 2c,d)

for . The maximal reaction rate K1dm is then given by

and the gain is , which proves that active transport can optimize reactivity as in higher dimensions. Interestingly, the c dependence of the gain is much more important than for d=2,3, which shows that the efficiency of active transport is strongly enhanced in one-dimensional or nematic structures at low concentration. Indeed, with the same values of D, v and a as given above for a vesicle in standard cellular conditions, and for a typical distance between reactants b≳100 μm (such as low concentrations of axonal receptors), we obtain a typical gain G1d≳100 (see Fig. 2c,d). In the limit of finite reactivity (k finite and D→0), we have and the same optimal value equation (2) of τ2,1dopt. As in higher dimensions, τ2,1dopt depends neither on the thermal diffusion coefficient D of phases 1, nor on the association constant k, which shows that the optimal interaction time with motors τ2opt presents remarkable universal features. Furthermore, our approach permits an estimate of τ2opt compatible with observations in standard cellular conditions, which suggests that cellular transport could be close to optimum.

Methods

The approximation scheme to solve the integro-differential equations (1) relies on the auxiliary function

and on the following decoupling hypothesis:

Similar arguments as provided in ref. 23 then lead to the diffusion-like equation

where . After rewriting (1) as

equations (3) and (4) provide a closed system of linear differential equations for the variables s and t1, whose resolution is tedious but standard. For d=3, we obtain in the limit of perfect reaction and low density a≪b:

where and T=tanh(α2a). This decoupling assumption has been controlled numerically for a wide range of parameters for d=2 (ref. 23), and is shown here to also be satisfactory for d=3 (Fig. 2a,b). For d=1, the decoupling approximation is exact and yields after straightforward calculations an explicit though hardly handleable form of the reaction time. In the regime D/v≪a≪b, we obtain in the limit the simple form

where . Optimization is then carried out using standard methods of functional analysis.