Abstract
The cell cytoskeleton is a striking example of an ‘active’ medium driven outofequilibrium by ATP hydrolysis^{1}. Such activity has been shown to have a spectacular impact on the mechanical and rheological properties of the cellular medium^{2,3,4,5,6,7,8,9,10}, as well as on its transport properties^{11,12,13,14}: a generic tracer particle freely diffuses as in a standard equilibrium medium, but also intermittently binds with random interaction times to motor proteins, which perform active ballistic excursions along cytoskeletal filaments. Here, we propose an analytical model of transportlimited reactions in active media, and show quantitatively how active transport can enhance reactivity for large enough tracers such as vesicles. We derive analytically the average interaction time with motor proteins that optimizes the reaction rate, and reveal remarkable universal features of the optimal configuration. We discuss why active transport may be beneficial in various biological examples: cell cytoskeleton, membranes and lamellipodia, and tubular structures such as axons^{1}.
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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 Factin and microtubules) of the cytoskeleton^{1}. 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 proteins^{1,15,16}. Active transport in cells has been extensively studied both experimentally, for instance by singleparticle tracking methods^{11,12}, and theoretically by evaluating the mean displacement of a tracer^{13,17}, or stationary concentration profiles^{14}.
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 tackled^{18}. Here, we propose an analytical model that enables us to determine for the first time the kinetic constant of transportlimited reactions in active media.
The model relies on the idea of intermittent dynamics introduced in the context of search processes^{19,20,21,22,23,24,25,26,27}. We consider a tracer particle evolving in a ddimensional 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 ω_{v} (Fig. 1a). The distribution of the filaments’ orientation is denoted by ρ(ω_{v}), and will be taken as either disordered or polarized (Fig. 1a), which schematically reproduces the different states of the cytoskeleton^{1}. The random duration of each phase i is assumed to be exponentially distributed with mean τ_{i}. The tracer T can react with reactants R (supposed immobile) only during freediffusion 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.
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 meanfield approximation of the general case of randomly located reactants with concentration c=a^{d}/b^{d}, where b is the typical distance between reactants. We start from a meanfield approximation of the firstorder reaction constant^{28} and write K=1/〈t〉, where the key quantity of our approach is the reaction time 〈t〉, which is defined as the mean firstpassage time^{29,30} (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 equation^{29}:
where t_{1} is the MFPT starting in phase 1 at position r, and t_{2} is the MFPT starting in phase 2 at position r with velocity v. I_{a} is the indicator function of the ball of radius a. As these equations (1) are of integrodifferential type, standard methods of resolution are not available for a general distribution ρ.
However, in the case of a disordered distribution of filaments (ρ(ω_{v})=1/Ω_{d}, where Ω_{d} is the solid angle of the ddimensional 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 K_{3d} is given in the Methods section, and is plotted in Fig. 2a,b. Strikingly, K_{3d} can be maximized (Fig. 2a,b) as soon as the reaction radius exceeds a threshold a_{c}≃D/v for the following value of the mean interaction time with motors:
where x_{0} is the solution of 2tanh(x)−2x+xtanh(x)^{2}=0. The τ_{1} dependence is very weak, but we can roughly estimate the optimal value by τ_{1,3d}^{opt}≃6D/v^{2}. This in turn gives the maximal reaction rate
so that the gain with respect to the reaction rate K_{3d}^{p} in a passive medium is G_{3d}=K_{3d}^{m}/K_{3d}^{p}≃C a v/D with C≃0.26.
Several comments are in order. (1) τ_{2,3d}^{opt} 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 a_{c} should be discussed. In standard cellular conditions, D ranges from ≃10^{−2} μm^{2} s^{−1} for vesicles to ≃10 μm^{2} 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 cargo^{1}. This gives a critical reaction radius a_{c} 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 threedimensional disordered case, active transport can optimize reactivity for sufficiently large tracers such as vesicles, as motormediated 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 motormediated motion. This could help justify that many molecular species in cells are transported in vesicles. Interestingly, in standard cellular conditions τ_{2,3d}^{opt} 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 observations^{1}, 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 G_{3d}≳2.5 (Fig. 2a,b) and can reach G_{3d}≳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 lowerdimensional 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 cells^{1}. 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 K_{2d} can be optimized in the regime D/v≪a≪b. Remarkably, the optimal interaction time τ_{2,2d}^{opt} takes the same value in the two limits and D→0 :
which indicates that again τ_{2,2d}^{opt} does not depend on the parameters of the thermal diffusion phase, neither through D nor k. In the limit , we have τ_{1,2d}^{opt}=(D/2v^{2})(ln^{2}(1/c^{1/2})/(2ln(1/c^{1/2})−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 a_{c}≃D/v of the same order as in the d=3 case) such as vesicles. However, here the gain G_{2d} 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 G_{2d}≳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 cell^{1}, can be shown to be equivalent in a first approximation to the onedimensional case, which is exactly solvable (Fig. 1a,b). The d=1 case is also important on its own in cell biology, as many onedimensional active structures such as axons, dendrites or stress fibres^{1} 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 τ_{2} (see Fig. 1b). For this type of tracer, the MFPT satisfies equations (1) with an effective nematic distribution of filaments ρ(ω_{v})=(1/2)(δ(v−e)+δ(v+e)). The reaction rate K_{1d} 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 K_{1d}^{m} 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 onedimensional 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 G_{1d}≳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 τ_{2,1d}^{opt}. As in higher dimensions, τ_{2,1d}^{opt} 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 τ_{2}^{opt} presents remarkable universal features. Furthermore, our approach permits an estimate of τ_{2}^{opt} compatible with observations in standard cellular conditions, which suggests that cellular transport could be close to optimum.
Methods
The approximation scheme to solve the integrodifferential equations (1) relies on the auxiliary function
and on the following decoupling hypothesis:
Similar arguments as provided in ref. 23 then lead to the diffusionlike equation
where . After rewriting (1) as
equations (3) and (4) provide a closed system of linear differential equations for the variables s and t_{1}, 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(α_{2}a). 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.
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Loverdo, C., Bénichou, O., Moreau, M. et al. Enhanced reaction kinetics in biological cells. Nature Phys 4, 134–137 (2008). https://doi.org/10.1038/nphys830
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DOI: https://doi.org/10.1038/nphys830
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