Protein Machines As Force Dipoles

Molecular machines are biomolecules, most often proteins, that undergo structural changes in shape during their operation cycles. These cyclic shape changes, induced by ligand binding and product release, take place under nonequilibrium conditions; therefore, they differ from thermally induced shape fluctuations for which microscopic reversibility holds and the fluctuation-dissipation theorem applies. These molecular machines operate in a viscous environment and their dynamics takes place under low Reynolds number conditions so that inertia does not play a significant role. As a result, if a force is applied to a particle in the fluid, the same force acts on the fluid.

Such protein machines act as stochastic oscillating force dipoles that can influence the motions of other particles in the system. For example, consider a protein with two domains that operates as an enzyme converting substrate into product molecules. The protein domains close in response to binding of ATP or other substrate molecules and open after the reaction and release of a product. We assume that the substrate is continuously supplied and the products are instantaneously removed from the system. When the substrate binds to the protein new bonds are formed and thus the chemical energy, needed to induce conformational changes and cause the domains to close, is supplied. When the product is released, the additional chemical bonds are broken, leading to domain opening, and the protein returns to its initial state. Within one cycle, an active protein consumes the chemical energy whose net value is determined by the difference in internal energies of the substrate and product molecules. If reverse conversion of a product into the substrate is allowed, an active protein can also operate in the opposite direction. Generally, its cycles are driven by the difference in Gibbs potentials of substrate and product; the sign of this difference determines the operation direction of the machine. Because the net force on the protein is zero, the forces that act on the domains are equal in magnitude and opposite in direction, so that a force dipole is created. This oscillating force dipole will act on the surrounding viscous fluid to generate hydrodynamic flows that can induce motions of passive particles in the fluid. A simple dimer model of such an active protein, where the domains in a bidomain protein are represented by beads, is formulated in SI Text.

Hydrodynamic Effects

When a force F is applied to the fluid at a point r, it produces a fluid flow field at R that advects a particle at this location with velocityR˙α=Gαβ(R−r)Fβ(r),[1]where Gαβ is the mobility tensor which, for sufficiently large distances, can be evaluated in the Oseen approximation. (The Einstein summation convention over repeated indices will be used throughout this paper.) For an oscillating dimer of length x with orientation given by the unit vector e and interaction force magnitude F, we have R˙α=[Gαβ(R−r−xe)−Gαβ(R−r)]eβF, because the forces on the dimer beads have equal magnitude but are opposite in direction. If the dimer length x is small compared with the distance |R−r|, we can approximately writeR˙α=∂Gαβ∂rγeβeγm,[2]where m(t)=x(t)F(t) denotes the magnitude of the force dipole. Although we derived Eq. 2 for a specific two-bead model, it is general. If an object immersed in the fluid changes its shape, it generates a hydrodynamic flow that, at large separations from this object, can be described as being produced by an active force dipole, unless special symmetries are present. Below, we shall treat any active protein machine as a nonequilibrium stochastic force dipole.

Consider a collection of such active force dipoles, located at positions {Ri} with orientations {ei} and subject to fluctuations arising from thermal and active nonthermal effects. The oscillating dipoles collectively create a fluctuating flow field that induces stochastic advection of a passive particle. At low Reynolds numbers, the passive particle will exactly follow the local flow velocity field. The equation of motion of a passive particle at point Rα may then be written asR˙α=∑i∂Gαβ(R−Ri)∂Riγeiβeiγmi(t)+fα(t).[3]The passive particle is subject to fluctuations from two sources. The random force fα(t) has zero mean and satisfies the fluctuation-dissipation relation, 〈fα(t)fα′(t′)〉n=2kBTγδαα′δ(t−t′), where γ is the mobility coefficient of the passive particle. The particle is also subject to thermal and active nonthermal fluctuations of the force dipoles mi(t) with zero mean, 〈m(t)〉n=0, and correlation function 〈mi(t)mj(0)〉n=〈m(t)m(0)〉nδij=S(t)δij, which defines the force dipole correlation function S(t)=〈m(t)m(0)〉n. For simplicity we have assumed that the force dipole correlations of different proteins are independent. We have further assumed that their orientations are randomly distributed. The force dipole tensor eiβeiγmi(t) of a protein is invariant under inversion of the orientation vector ei and has properties like the tensor order parameter in nematic liquid crystals. When correlations among active proteins are taken into account it may be possible that nematically ordered active protein states could be found (26, 36). We shall not consider such effects here.

The force dipole correlation function S(t) plays a central role in our study. This correlation function contains effects that arise from both thermal and nonthermal noise. This correlation function may be written as S(t)=2STδ(t)+SA(t), where the thermal component ST(t) is determined by the temperature, whereas the active component SA(t) results from nonthermal colored noise. In the absence of substrate, or when the system operates under equilibrium conditions, only the delta correlated thermal contribution remains. The integral intensity S=∫0∞dt S(t)≡ST+SA may also be defined. This quantity will enter in the expressions for the diffusion and drift derived below.

Because the conformational transitions that produce the force dipole depend on substrate binding, the dependence on substrate concentration enters the description through the force dipole correlation function S(t) and, thus, the integral intensity SA of active force dipoles should depend on the substrate concentration cS. Although the precise form of the dependence SA(cS) can only be determined by considering the kinetics of a particular enzymatic molecular system, general comments on its structure may nonetheless be made. The contribution SA of active force dipoles must vanish in the absence of substrate because only thermal fluctuations are then present and they are accounted for in ST. At high enough substrate concentrations the enzymes will operate at their maximum conversion rates and nonthermal effects will saturate. Consequently, SA(cS) should display a similar behavior: a linear proportionality at small substrate concentration cS and saturation at large concentrations. For example, a functional form like that of Michaelis–Menten kinetics, SA(cS)∼S0cS/(K+cS), where S0 is the saturation value of SA and K is a constant, satisfies these criteria, but other forms for SA(cS) with similar limiting regimes may well result from a calculation of SA for a specific molecular system.

If, within the time interval being considered, the displacements in the position R(t) of a passive particle are small, we can write Rα(t)=R0,α+ρα(t) and retain only terms that are of the first order in the displacements ρα(t), so thatρ˙α=∑i[∂Gαβ(R0−Ri)∂Ri,γ+∂2Gαβ(R0−Ri)∂Ri,γ∂R0,δρδ]×ei,βei,γmi(t)+fα(t).[4]It is convenient to write the first term on the right side of this equation in field-point notation,ρ˙α=∫dr [∂Gαβ(r)∂rγ−∂2Gαβ(r)∂rγ∂rδρδ]×∑iei,βei,γmi(t)δ(Ri−R0−r)+fα(t),[5]where the term involving the sum over the proteins represents the microscopic density of force dipoles, ei,βei,γmi(t), at a point r with origin at the position R0 of the passive particle. Note that the first and the second derivatives of the Green function correspond to dipole and quadrupole contributions. This equation shows that the instantaneous position of a passive particle evolves with time according to a stochastic differential equation with nonthermal multiplicative noise arising from the collective operation of active force dipoles, in addition to the additive thermal noise.

The diffusion tensor Dαα′ and the mean drift velocity V¯ of a passive particle can be determined fromDαα′=∫0∞dt 〈δVα(t)δVα′(0)〉, V¯α=〈Vα〉,[6]where δVα=Vα−V¯α and the angle bracket 〈⋯〉 denotes an average over the stochastic fluctuations, both thermal and nonthermal, as well as the orientations and positions of active force dipoles.

The diffusion tensor and mean drift velocity of the passive particle may be obtained by substituting the expression in Eq. 5 for the velocity Vα=ρ˙α of a particle, retaining only leading terms, into Eq. 6. As discussed earlier, when computing the average values in Eq. 6, we assume that the orientations of active force dipoles are not correlated with their positions so that 〈∑ieβeβ′eγeγ′δ(Ri−R0−r)〉=〈eβeβ′eγeγ′〉c(r)≡Ωββ′γγ′c(r), where Ωββ′γγ′=Cd[δββ′δγγ′+δβγδβ′γ′+δβγ′δβ′γ], with C2=1/8 and C3=1/15 for two and three dimensions, respectively, and c(r)=〈∑iδ(Ri−r)〉 is the local concentration of active force dipoles at a point r in the fluid. We findDαα′(R0)=Dαα′T(R0)+SAΩββ′γγ′∫dr ∂Gαβ(r)∂rγ∂Gα′β′(r)∂rγ′c(R0+r)≡Dαα′T(R0)+Dαα′A(R0)[7]V¯α(R0)=−SAΩββ′γγ′∫dr ∂2Gαβ(r)∂rγ∂rδ∂Gδβ′(r)∂rγ′c(R0+r),[8]where Dαα′T(R0) is the equilibrium diffusion tensor of the passive particle averaged over protein configurations. It contains effects arising from the thermal contribution ST as well as the mobility of the individual passive particle. The last line in Eq. 7 defines the contribution of active force dipoles, Dαα′A(R0), to the total diffusion tensor.

We shall now analyze Eqs. 7 and 8 separately for 3D and 2D systems.

Numerical Estimates

The magnitude of a force dipole m of a protein machine can be roughly estimated as m∼FdP, where F is the force generated by the machine and dP is the linear size of the protein. Molecular motors, such as myosin or kinesin, typically generate forces about 1 pN and this can be chosen as the characteristic value for F. Taking the size of a protein to be about 10 nm, the force dipole can be estimated to be about m=10−20 N⋅m. The correlation time for force-dipole fluctuations can be taken to be the duration tc of the cycle time in a chemical machine. Although enzyme cycle times vary widely, we choose a time of about tc∼1 ms. The parameter SA can then be evaluated to give SA∼m2tc=10−43 N2⋅m2⋅s. Note that this estimate corresponds to substrate (typically ATP) saturation conditions: The machine binds a new substrate molecule and enters into a new cycle immediately once the previous cycle finishes. If this condition is not satisfied, the protein machine must wait for a new substrate molecule to arrive. During this waiting period, the machine does not act as a force dipole and this will decrease the value of SA. Obviously, the effects disappear when the substrate is not supplied.

Concentrations of active proteins inside a biological cell can vary over a large range. The highest concentrations of the order of 10−4 M are characteristic for the enzymes involved in glycolysis. As a rough estimate, a value of 10−6 M can be chosen, corresponding to c0=1021m−3, so that the mean distance between the proteins being considered is about 100 nm. Given the protein size, we choose a cutoff length of ℓc=10 nm. The viscosity of water is about 10−3 Pa⋅s. For the dimensionless numerical factors in Eqs. 11 and 14 we take their order-of-magnitude values ζ3=10−2 and ξ3=10−2.

With these values, the contribution (Eq. 10) to the diffusion coefficient owing to hydrodynamic effects arising from protein machines in bulk 3D solutions is about DA≈10−6cm2/s. This result should be compared with typical diffusion constants in water that can vary from about 10−5cm2/s for small molecules to 10−7cm2/s for small proteins in water. If one takes the viscosity of the cytoplasm to be two to four times that of water, the active contribution DA will decrease by approximately an order of magnitude. However, it is still approximately the same as the thermal Brownian contribution DT≈kBTγ=kBT/(6πηRp), where Rp is the radius of the passive particle, under the same viscosity conditions using a Stokes law estimate. To estimate the magnitude of the drift velocity from Eq. 13, we can take ∇c=Δc/L, where Δc is the concentration difference across the cell and L=10 μm is the typical eukaryotic cell size. If we choose Δc≈0.1c0, where again c0=1021m−3, we obtain a drift velocity magnitude of about V¯≈1 μm/s.

Proceeding to lipid bilayers, we observe that their 3D viscosity ηL∼1 Pa⋅s is typically a factor of 103 higher than that of water. The 2D viscosity of such bilayers is ηm=ηLh, where h∼1 nm has been taken as the thickness of the bilayer. The magnitude S∼10−43 N2⋅m2⋅s may again be used for the force dipoles. Taking the mean distance between proteins to be ∼100 nm, the 2D concentration is about c0=1014 m−2. As rough estimates of these factors we again choose their order-of-magnitude values, ζ2=ξ2=10−2. A membrane of micrometer size is considered and a cut-off distance of 10 nm is introduced.

Given these numerical values, the hydrodynamic effects of active protein inclusions are predicted to increase the diffusion of passive particles within the membrane by about DA≈10−9cm2/s. For comparison, Brownian diffusion constants for proteins in lipid bilayers are of the order of 10−10cm2/s and diffusion constants for lipids are about 10−8cm2/s. The characteristic magnitude of the drift velocity of passive particles in lipid bilayers is estimated to be about V¯≈10−2 μm/s, assuming that Δc=0.1c0 and the characteristic length for concentration variation in a membrane is about 1 μm.

These numerical estimates should be used only as rough guide to the possible magnitudes of the effects because many of the parameters may vary significantly from one system to another, or are known only poorly. For example, forces have only been measured for some molecular motors and, for protein machines that are not motors or for enzymes, they may be smaller than 1 pN. However, the concentrations of some proteins that behave as force dipoles may be significantly higher than the value we have assumed. For example, the enzyme phosphoglycerate kinase involved in glycolysis is present in the living cell in the concentrations up to 10−4 M, two orders of magnitude larger than our assumed value. Proteins typically contribute about 40% of the mass in biomembranes and their concentrations may well be significantly higher than our 2D values. In addition, the cytoplasm is a very complicated medium that is crowded by a variety of macromolecules, filaments, organelles, and other structures (37). Diffusion in such a crowded environment differs from that in a simple solution. Crowding can also influence the hydrodynamic effects owing to active force dipoles discussed in this paper. Such effects will have to be considered in descriptions of transport in the cell that model the detailed structure of the cytoplasm. Consequently, the uncertainty in the estimates of hydrodynamic effects is high and deviations of up to two orders of magnitude from the numerical estimates given above may well be possible.

The effects considered here depend on the concentration of substrate through SA(cS) as discussed earlier. If the substrate concentration varies in a cell or a membrane, the coordinate-dependent dipole intensity SA should be retained within integrals in Eqs. 7 and 8. In three dimensions, diffusion is determined by the local dipole force intensity according to Eq. 10, whereas the drift velocity is determined by the local gradient of SA(r)c(r), replacing the gradient of concentration in Eq. 13. In two dimensions, local diffusion and drift are generally dependent on the concentration distribution of active inclusions and substrates over the entire membrane.

Discussion and Conclusions

In a biological cell there are large populations of active proteins, both molecular machines and enzymes, that change their conformations within catalytic cycles. In this paper we showed that when active proteins are present, either in solution or in lipid bilayers, they can substantially modify the diffusion constants of passive particles in the system. These modifications affect all passive particles, and all active proteins, even of different kinds, contribute to the effect provided they are supplied with substrate and remain active. The magnitude of the effect can be comparable to the value of Brownian diffusion constants under physiological conditions.

Furthermore, if protein machines are nonuniformly distributed in a cell or in a biomembrane, directed drift of passive particles, analogous to chemotaxis, can occur. However, the mechanism is completely different: All active proteins contribute toward it and all passive particles experience the drift. Drift velocities of the order of micrometers per second can be realized. The enhancement of diffusion and chemotaxis-like drift should take place for the protein machines (enzymes) themselves as well. Note that the drift velocity is in the same direction as the concentration gradient and therefore the hydrodynamic attraction of incoherent active proteins should occur. Generally, the same proteins would exhibit different interactions depending on whether they are catalytically active or inactive (no ATP is supplied). However, thus far we have not considered collective effects due to hydrodynamic interactions on the populations of active proteins. It may be that orientation alignment leading to nematic order (26, 36) and cycle synchronization also arise.

In three dimensions, hydrodynamic interactions are already long-ranged, with power-law dependence. In two dimensions they become ultra-long-ranged with a logarithmic dependence on distance. Thus, the effects predicted to exist in 3D and 2D systems differ substantially. In solution, the change in the diffusion constant is determined by the local protein machine concentration and the drift velocity is controlled by its local spatial gradient. In contrast, in 2D systems such as lipid bilayers, the effects are essentially nonlocal: The change in the diffusion constant and the drift of passive particles at a given location are determined by the distribution of active inclusions over the entire membrane. Note, however, that only relatively small membranes with micrometer sizes were considered here. In general, diffusion in biomembranes should be anisotropic, reflecting the asymmetry of protein distribution and the membrane shape.

Our description of how active molecular machines, through hydrodynamic interactions, influence the dynamics of passive particles was based on the equations of continuum hydrodynamics. One might question the use of such a continuum description for molecular systems. It is well established (38) that hydrodynamic effects are observable on even very small scales of tens of solvent particle diameters. They persist despite strong fluctuations and their presence is signaled in the long-time tails of velocity correlation functions or even in the transport properties of polymers. Our use of continuum equations is restricted to rather long scales so that the main conclusions of our study should be robust.

Sen and coworkers (39, 40) have shown that catalytically active enzymes have larger diffusion coefficients than their inactive counterparts in the absence of substrate. Recently, chemotaxis-like drift of enzymes in the presence of substrate gradients has been observed and used for sorting of the enzymes (41). Although additional analysis of the experimental data is needed, it may be that such observations can be explained by the effects considered above. Furthermore, in vivo studies have revealed that the diffusion of particles decreases in ATP-depleted biological cells (42). In addition, several studies have proposed specific explanations for the importance of nonthermal random motions in living cells, whose origin lies in the forces generated by the uncorrelated activity of protein machines. (26, 42⇓–44) Such nonthermal fluctuations may also be a consequence of the universal hydrodynamic effects, described here, that arise from active conformational changes in molecular motors and other protein machines powered by ATP.

Our analysis focused on general aspects of the phenomena and is not intended to address the full complexity seen in biological systems. Nevertheless, in accord with the above findings, our results suggest a modified physical picture of kinetic processes in the biological cell. When energy is supplied by ATP or other substrates to active proteins in a cell, such as molecular motors, other protein machines, or enzymes, they cyclically change their shapes in the course of carrying out their various specific functions. In addition to their functions, all such proteins act as oscillating active force dipoles and collectively create a fluctuating hydrodynamic field over the entire cell or a biomembrane. This nonequilibrium flow field can be maintained because a fraction of the energy flux arriving with substrates is diverted through the force-dipole activity to hydrodynamic flows in the cytoplasm. Such fluctuating fields arise from nonequilibrium effects; therefore, in contrast to thermal hydrodynamic fluctuations, the fluctuation-dissipation theorem does not apply to them. Because these fluctuating fields arise from nonthermal noise, it is possible that they can be rectified and work or energy can be extracted from them. Thus, active proteins in a cell not only execute their specific functions but, collectively, they supply power in a distributed way to the system. Such power, originating from substrate supply to active proteins, spans the entire cell.