Diffusion of an Enzyme: the Role of Fluctuation-Induced Hydrodynamic Coupling
DDiffusion of an Enzyme: the Role of Fluctuation-Induced Hydrodynamic Coupling
Pierre Illien,
1, 2
Tunrayo Adeleke-Larodo, and Ramin Golestanian Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK Department of Chemistry, The Pennsylvania State University, University Park, PA 16802, USA (Dated: October 13, 2017)The effect of conformational fluctuations of modular macromolecules, such as enzymes, on theirdiffusion properties is addressed using a simple generic model of an asymmetric dumbbell madeof two hydrodynamically coupled subunits. It is shown that equilibrium fluctuations can lead toan interplay between the internal and the external degrees of freedom and give rise to negativecontributions to the overall diffusion coefficient. Considering that this model enzyme explores amechanochemical cycle, we show how substrate binding and unbinding affects its internal fluctua-tions, and how this can result in an enhancement of the overall diffusion coefficient of the molecule.These theoretical predictions are successfully confronted with recent measurements of enzyme dif-fusion in dilute conditions using fluorescence correlation spectroscopy.
PACS numbers: 05.40.-a, 82.39.-k, 47.63.mf
Introduction.—
The highly precise and efficient func-tions performed in a biological cell, such as vesiculartransport or DNA synthesis, require the conversion ofchemical energy into mechanical work by biomolecules[1–3]. To this purpose, enzymes and motor proteins per-form cyclic turnovers in which they bind to substratemolecules and catalytically convert them to productswhile undergoing conformational changes, which affecttheir transport and diffusion properties. Therefore, thequestion of whether a biological molecule is able to pro-duce enough mechanical work to overcome the thermalfluctuations of its environment is central for the under-standing of biological self-organization and intracellulartransport [4–11].In this context, fluorescence correlation spectroscopy(FCS) has proven to be a powerful tool to study thephysical properties of macromolecules, such as the fold-ing/unfolding or denaturation dynamics of proteins [12,13]. Recently, in vitro studies of dilute solutions of en-zyme molecules using FCS have revealed that their dif-fusion coefficient is enhanced when they are catalyticallyactive [14–17]. This phenomenon may contribute to theself-organization of biological processes such as the Krebscycle [18]. The experimental observation holds for a widerange of enzymes with very different kinetic and ther-modynamic properties. Although it was suggested thatdiffusion enhancement could be correlated to the exother-micity of the reaction catalyzed by the enzyme or its over-all catalytic rate [17, 19–21], we have recently shown thatthe slow and endothermic enzyme aldolase could exhibita similar behaviour [22]. The new observations cannotbe theoretically explained within the existing nonequi-librium paradigm, and a completely new approach is re-quired. In this Letter, we use a new paradigm to providea quantitative description for this phenomenon.
Main results.—
We propose a simplified descriptionfor a generic macromolecular complex using the modelof an asymmetric dumbbell, which represents the modu- lar structure of the macromolecule and which reduces itsinternal degrees of freedom to a minimal number [Fig.1(a)]. Considering the hydrodynamic interactions be-tween the different parts of the enzyme, we show howthe internal degrees of freedom affect its overall diffu-sion. More precisely, we show that the effective diffusioncoefficient of the dumbbell has the generic form D eff = D ave − δD fluc , (1)where the first term corresponds to the thermal aver-age of the contributions due to the translational modesof the dumbbell, whereas the second term representsfluctuation-induced corrections arising from the internalelongation and rotation degrees of freedom. The negativesign of the correction term, which is controlled by theasymmetry of the dumbbell of the individual subunits, isa generic feature of fluctuation-induced interactions [23].We then propose a simplified description of themechanochemical cycle visited by the enzyme [Fig. 1(b)],and its influence on the fluctuations of the enzyme. Sub-strate or product binding generically hampers the fluctu-ations of the modular structure [Figs. 1(c) and 1(d)], andtherefore reduces the fluctuation-induced contribution tothe effective diffusion coefficient. We find that this leadsto a relative enhancement of the diffusion ceofficient witha generic dependence on the substrate concentration S of the form∆ DD ≡ D ( S ) − D ( S = 0) D ( S = 0) = A · S S + K , (2)where K is the effective equilibrium constant of the chem-ical cycle, and A a numerical prefactor that depends onthe geometrical and physical properties of the enzyme. Model.—
Our goal is to investigate the role played bysolvent-mediated hydrodynamic interactions between thedifferent parts that constitute the model enzyme. Realmacromolecules generally have a very large number of a r X i v : . [ c ond - m a t . s o f t ] O c t S P (a) (b)(c) (d) + + x R x x a û û û û FIG. 1. (a) Generalized dumbbell model: two subunits, which represent the modular structure of the enzyme, interact viahydrodynamic interactions and a harmonic-like potential U . The conformation of the enzyme is described by the positions ofthe subunits x α and their orientations ˆ u α . (b) Three-state mechanochemical cycle explored by the enzyme in the presence ofsubstrate and product molecules: when it is free, the enzyme can bind to a substrate molecule and transform it into a productmolecule which is ultimately released in the bulk. These transformations are assumed to be reversible. (c) Modification of theextent of elongation fluctuations of the dumbbell due to substrate or product binding. (d) Modification of the orientationalfluctuations upon substrate or product binding. internal degrees of freedom. In order to describe their in-ner dynamics, different types of fluctuation modes are tobe considered, among which compressional modes, thatcome from the protein elastic properties, and orienta-tional modes, that originate from the hinge motion offreely rotating subparts. To study these two types ofinternal degrees of freedom, and in line with previousmodels of low Reynolds number swimmers [24–26], wereduce the complex geometry of the enzyme to a sim-ple model molecule taking the form a generalized dumb-bell made of two subunits, which are geometrically dif-ferent and whose shapes reproduce the modular struc-ture of the enzyme. They interact via hydrodynamicinteractions and via a harmonic-like potential U of stiff-ness k and equilibrium distance a , with a short-distancecutoff that accounts for steric constraints. Their posi-tions x α and orientations ˆ u α undergo thermal fluctua-tions [Fig. 1(a)]. This simplified model does not aimto represent a specific enzyme, but will allow us to carryout a detailed analytical study of the fluctuation-inducedeffects that arise from the hydrodynamic coupling. Al-though this model can be related to previous attemptsto use hydrodynamically-coupled dumbbells or chains todescribe the diffusion of polymers [27], the question ofthe internal asymmetries of the dumbbell and of therole played by orientational fluctuations was generallyleft aside, and our approach offers a novel insight in thefluctuation-induced hydrodynamic coupling between thedumbbell subunits. Smoluchowski description.—
The starting point of our analytical treatment is the Smoluchowski equationobeyed by P ( x , x , ˆ u , ˆ u ; t ), namely, the probabilityto find subunit α at position x α and with orientation ˆ u α [Fig. 1(a)] at time t : ∂ t P = (cid:88) α,β =1 , (cid:110) ∇ α · M αβ TT · [( ∇ β U ) P + k B T ∇ β P ]+ ∇ α · M αβ TR · [( R β U ) P + k B T R β P ]+ R α · M αβ RT · [( ∇ β U ) P + k B T ∇ β P ]+ R α · M αβ RR · [( R β U ) P + k B T R β P ] (cid:111) , (3)where the hydrodynamic tensors M αβ AB describe the in-teractions between translational (T) and rotational (R)modes of the subunits [28, 29], U ( x , x , ˆ u , ˆ u ) is theinteraction potential between the subunits, and the ro-tational gradient operator is R α ≡ ˆ u α × ∂ ˆ u α [27]. Weconveniently rewrite Eq. (3) using the center of mass R = ( x + x ) / x = x − x = x ˆ n coordinates. We will use the followingtype of approximation for any combination of the mobil-ity tensors: A (cid:39) a . This approximation correspondsto the usual pre-averaging of mobility tensors, whichconsists in neglecting any off-diagonal terms [30], andwhich has been widely used in polymer physics, and isknown to have a wide range of validity [27]. The effect ofthe subunits anisotropy, and therefore of the orientation-dependence of the mobility tensors will be the subjectof a later publication. The orientation-dependence ofthe potential U is assumed to take the simple form U = k ( x − a ) [1 + (cid:80) α =1 , V α ˆ n · ˆ u α + V ˆ u · ˆ u ], wherethe dimensionless constants V α and V characterize thestrength of the constraints on orientational fluctuations,and vanish for freely rotating subunits.Our aim is to calculate the effective centerof mass diffusion coefficient defined as D eff =lim t →∞
16 dd t (cid:82) R (cid:82) x (cid:82) ˆ u (cid:82) ˆ u R P . A direct attempt to ob-tain an evolution equation for the second moment (cid:10) R (cid:11) from Eq. (3) yields an unclosed equation, which includescorrelation functions that involve both the external ( R )and internal ( x , ˆ u and ˆ u ) degrees of freedom. Us-ing Eq. (3) to write the evolution equation of thesequantities, one can show that it involves higher-ordercorrelation functions; therefore, we need an approxima-tion to close this hierarchy of equations. We next iden-tify the hierarchy of the characteristic time scales asso-ciated with the different degrees of freedom: from slowto fast we have the center of mass position R , the ori-entations ˆ n , ˆ u and ˆ u , and the elongation x . Theelongation relaxation time is indeed τ s = ξ/k , where ξ is an estimate of the friction coefficient of the en-zyme, whereas the rotational diffusion time of the en-zyme τ r , such that (cid:104) ˆ n (0) · ˆ n ( t ) (cid:105) ∼ exp( −| t | /τ r ), is ofthe order of ξa /k B T . Forming the ratio between thesetwo characteristic times yields the dimensionless num-ber τ s /τ r ∼ k B T /ka ∼ δx/a , which is a measure of therelative deformation of the molecule due to thermal fluc-tuations, and which is smaller than unity.Guided by this ordering, we average Eq. (3) over theradial coordinate x assuming that the orientations ˆ n , ˆ u and ˆ u of the dumbbell are fixed. We define the average (cid:104)·(cid:105) = P (cid:82) d x x · P where P = (cid:82) d x x P . The result-ing equation satisfied by P ( R , ˆ n , ˆ u , ˆ u ; t ) is presentedin the Supplementary Information [31]. The next stepof the calculation consists of a moment expansion of theequation satisfied by P with respect to the orientationvectors ˆ n , ˆ u and ˆ u , which yields another hierarchy ofequations that will need to be approximated by usinga closure scheme for orientational order parameters [32–34]. This calculation gives the following leading orderexpression for the effective diffusion coefficient [31] D eff = k B T (cid:104) m (cid:105)− k B T (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) (cid:34) − (cid:88) α =1 , (cid:18) ka k B T V α (cid:19) K α (cid:35) (4)where we defined the tensors M = M + M +2 M , Γ = M − M and W = M + M − M . Thisresult has the structure of Eq. (1), where the negativefluctuation-induced corrections are controlled by the co-efficient γ , which is a measure of the asymmetry of thedumbbell. K α is a dimensionless coefficient of order 1,that depends on the geometry of the dumbbell and thatis estimated to be positive for harmonic-like potentials[31]. Fluctuation-dissipation theorem.—
To validate theabove results, we verify that our treatment of the problemsatisfies the simplest fluctuation theorem, which takes theform of the usual Einstein relation µ eff = D eff /k B T [35],where µ eff is the mobility coefficient of the dumbbell.For simplicity, we do not consider here the effect of theorientation-dependence of the potential U . We assumethat each of the dumbbell subunits is submitted to anexternal force f / f / e where the amplitude f andthe unit vector ˆ u are arbitrary and constant, so that thetotal force on the dumbbell is f ˆ e . We aim to establishthe equation satisfied by the probability distribution of R and x in the presence of an external force and denotedby P f ( R , x ; t ). Rewriting Eq. (3) as ∂ t P = L T P in orderto define the equilibrium Fokker-Planck operator L T , wefind that P f satisfies ∂ t P f = L T P f − f ∇ R · [( M · ˆ e ) P f ] − f ∇ x · [( Γ · ˆ e ) P f ] . (5)Following the averaging procedure presented above,where one can integrate over the radial coordinate x assuming that the orientation is fixed, we first obtainthe equation satisfied by P f = (cid:82) d x x P f . Performingagain a moment expansion with respect to orientationof the equation satisfied by P f , we get a closed evo-lution equation of the density ρ f ≡ (cid:82) x P f as ∂ t ρ f = D eff ∇ ρ f − µ eff f · ∇ ρ f , and find that the effective mobil-ity coefficient satisfies the Einstein relation given above. Path-integral formulation.—
Given the set of approx-imations we used to close the hierarchy of equationsyielded by the orientation moment expansion, the abovecalculation only gives the long-time limit of the diffu-sion coefficient, and does not contain information aboutthe convergence to this asymptotic result. In order toget a better insight on the time dynamics of the general-ized dumbbell, we turn to a path-integral representationof the stochastic dynamics [36]. Ignoring for clarity theeffect of orientational fluctuations, the starting point isthe Langevin equation ˙ x αi = M αβ TT ,ij F βj + √ k B T σ αβij ξ βj where the contravariant indices (Greek letters) denotethe labels of the particles, and the covariant indices (Ro-man letters) correspond to the coordinates. The ten-sors σ αβ are defined as the “square root” of the mo-bility tensors and obey σ αγik σ βγjk = M αβ TT ,ij . The unitwhite noise ξ α satisfies (cid:104) ξ αi ( t ) (cid:105) = 0 and (cid:104) ξ αi ( t ) ξ βj ( t (cid:48) ) (cid:105) = δ ij δ αβ δ ( t − t (cid:48) ). The force F α is related to the potential U through F α = −∇ α U . The propagator conditioned onthe starting and arriving points of the dynamics is for-mally written as the integral of a constraint, and treatedfollowing the Martin-Siggia-Rose treatment of such path-integrals [37]. We obtain an expression in the form P ∝ (cid:82) (cid:81) α D x α ( τ ) exp {−S [ x ( τ ) , x ( τ )] } , where the ac-tion has the simple form S = 14 k B T (cid:90) d τ ( ˙ x α − M αγ TT F γ ) · Z αβ · ( ˙ x β − M βδ TT F δ ) . (6)Here, the force is F α = −∇ α U , and the friction tensors Z αβ are inverse to the mobility tensors.Using again the diagonal approximation A (cid:39) a aswell as the pre-averaging of the hydrodynamic tensors,and changing the variables in the path-integral in orderto use the coordinates x and R instead of x and x , weobtain the following action: S = 1 k B T (cid:90) d τ (cid:104) z (cid:105) R + S x , (7)with S x = 1 k B T (cid:90) d τ (cid:20) (cid:104) y (cid:105)
16 ˙ x + (cid:104) ζ (cid:105) x · ˙ R + (cid:104) w (cid:105) U (cid:48) (cid:21) , (8)where we define Z = Z + Z + 2 Z , Y = Z + Z − Z , and ζ = Z − Z . For a simple harmonic potential U = k x /
2, the x -dependent part of the action S x canbe written as the time integral of a quadratic form. Af-ter integration over the paths { x ( τ ) } , which correspondsto an integration of the fast degrees of freedom of thedumbbell, this yields P ∝ (cid:90) D R ( τ ) exp (cid:26) − k B T (cid:90) d ω π ω | R ( ω ) | µ ( ω ) (cid:27) , (9)where we have written the integral over τ in Fourierspace and defined a mobility as µ ( ω ) − = (cid:104) z (cid:105) − ( (cid:104) ζ (cid:105) ω / (cid:101) G ( ω ), with the Green’s function being (cid:101) G ( ω ) = ( (cid:104) y (cid:105) ω / (cid:104) w (cid:105) k ) − . We deduce the meansquare displacement of the center of mass using theGreen-Kubo relation [38], and find (cid:10) R (cid:11) = 6 k B T (cid:104) z (cid:105) (cid:34) t + (cid:104) ζ (cid:105) (cid:104) z (cid:105) (cid:104) y (cid:105) − (cid:104) ζ (cid:105) t (cid:63) (1 − e − t/t (cid:63) ) (cid:35) . (10)A time-dependent diffusion coefficient, defined as D eff ( t ) =
16 dd t (cid:10) R (cid:11) , is deduced, and relating the aver-aged resistance tensors (cid:104) z (cid:105) , (cid:104) y (cid:105) and (cid:104) ζ (cid:105) to the mobilitytensors [31], we find the following asymptotic regimes D eff = (cid:40) k B T (cid:104) m (cid:105) ; t (cid:28) t (cid:63) , k B T (cid:104) (cid:104) m (cid:105) − (cid:104) γ (cid:105) (cid:104) w (cid:105) (cid:105) ; t (cid:29) t (cid:63) , (11)where the crossover time t (cid:63) = ( (cid:104) w (cid:105) k ) − is the elon-gation relaxation time. The drop in the diffusion co-efficient between the two limiting regimes is therefore − δD fluc = − k B T (cid:104) γ (cid:105) (cid:104) w (cid:105) , which is in agreement with the re-sult obtained from the Smoluchowski description of thedynamics [Eq. (4)], where we showed the existence ofa negative correction to the diffusion coefficient of thedumbbell due to its asymmetry. We therefore confirmthis observation, and highlight the consistency betweenthe two treatments of the stochastic dynamics. The fulltime dependence of the effective diffusion coefficient isplotted in Fig. 2. . . . .
001 0 .
01 0 . D e ff ( t ) / D e ff ( ) t/t ? a /a = 1 a /a = 1 . a /a = 1 . FIG. 2. Time-dependence of the diffusion coefficient ofthe dumbbell as obtained by the path-integral formulation. D eff ( t ) is rescaled by its initial value D eff (0) and plottedagainst the rescaled time t/t (cid:63) (see text) for different valuesof the relative sizes of the protein subunits a and a , for a /a = 0 . k B T / ( ka ) = 0 .
01. For all plots, we comparethe results obtained with the mobility functions written inthe Oseen limit [29] (dashed lines) and for spherical subunitswith higher-order corrections [39] (solid lines).
Mechanochemical cycle.—
When the enzyme is placedin the presence of substrate molecules, it will go througha mechanochemical cycle: depending on whether the en-zyme is free, bound to a substrate molecule or bound to aproduct molecule, its conformation will fluctuate arounddifferent equilibrium states. In other words, substratebinding and unbinding can strongly affect the fluctua-tions of the internal degrees of freedom and thereforeimpact the overall diffusion coefficient of the enzyme, asunveiled by our simple model. In order to go further withthis idea, we consider the simplified three-state catalyticcycle represented in Fig. 1(a): the enzyme binds to a sub-strate molecule, and converts it into a product moleculethat is ultimately released. Both steps are assumed to bereversible.From a conformational point of view, the enzyme thenonly exists in two states where it is respectively free orbound, characterized by the potentials U f ( C ) and U b ( C ),where C is a high-dimensional vector describing the con-formation of the enzyme. It is straightforward to showthat, within this discrete-state equilibrium picture, theaverage of any conformation-dependent function writes (cid:104) Φ (cid:105) = (cid:104) Φ (cid:105) f + f ( S, P ) [ (cid:104) Φ (cid:105) b − (cid:104) Φ (cid:105) f ], where f is a functionof the substrate and product concentrations [31]: f ( S, P ) = SS + K S K P P + K S + PP + K P K S S + K P . (12)The effective equilibrium constants K S and K P are de-fined as K S = K S , (cid:82) C e − U f /k B T (cid:82) C e − U b /k B T , (13) K P = K P , (cid:82) C e − U f /k B T (cid:82) C e − U b /k B T , (14)where K S , and K P , are the bare equilibrium constantsof substrate and product binding respectively. We as-sume for simplicity that the chemical rates α ( S ) and δ ( P ) defined in Fig. 1(b) are linear functions of S and P respectively, with α ( S ) = α S and δ ( P ) = δ P . Thebare equilibrium constants then read K S , = β/α and K P , = γ/δ [31].We must emphasize that this simplified mechanochem-ical cycle neglects the nonequilibrium step of the reac-tion where substrate molecules are actually turned intoproduct molecules. The reversible binding and unbind-ing steps can indeed be assumed to happen faster thanthe chemical step [22, 40]. The equilibrium picture wepresent here is then valid at any stage of the chemicalreaction, whether the system is in a transient state orhas reached chemical equilibrium. Therefore, the expres-sion of the diffusion enhancement presented in Eq. (2) isvalid both at the early stages of the reaction, where veryfew substrate molecules have been converted into productand where the effective equilibrium constant is K = K S ,and at chemical equilibrium, where K = ( K S + K P ) / (cid:104) Φ (cid:105) b − (cid:104) Φ (cid:105) f ] / (cid:104) Φ (cid:105) f and stop at the lowest order,which will provide us with an expression of the form ofEq. (2) for the relative change in diffusion coefficient,where the dimensionless quantity A is a function of vari-ous mobility coefficients in the free and bound states, butnot the substrate concentration [31]. The simple expres-sion we obtain for ∆ D/D is to be compared with pre-vious experimental measurements. First, the Michaelis-Menten-like dependence on the total substrate concen-tration S corresponds to the experimental observationsthat we recently reported [22]. Secondly, since A is con-structed as a ratio between averages of similar quantities, . . .
15 1 10 100 1000 A s k b /k f a /a = 1 a /a = 1 . a /a = 1 . FIG. 3. The contribution to the amplitude A s when substratebinding modifies the stiffness of the interaction potential from k f to k b . In the free state, the potential stiffness is taken as k f = k B T /a . The mobility functions are written in the Oseenlimit [29] (dashed lines) and for spherical subunits of radii a and a with higher-order corrections [39] (solid lines). Wecompare different values of the asymmetry a /a , and theequilibrium length of the potential is such that a /a = 0 . by default we expect it to be of order unity, which is con-firmed by the measured relative diffusion enhancement A at substrate saturation ( S (cid:29) K ), found to be of theorder of a fraction of unity. Conformational changes.—
In order to be more spe-cific, we will finally investigate the consequences of anumber of typical structural modifications on the diffu-sion coefficient of the enzyme. First, taking the exampleof aldolase [40], binding will influence the average confor-mation by bringing the two subunits closer by a few ˚A.This can be incorporated by choosing U f = k f ( x − a ) ,and U b = k f [ x − ( a − δx )] , where a is the equilibriumdistance between the subunits (of the order of a few nm),and where δx represents the typical displacement of theprotein residues between different conformational states[Fig. 1(b)]. The substrate molecule may also play therole of a stiff cross-linker for the protein, and binding islikely to increase significantly the effective stiffness of theinteraction potential, that will read U b = k b ( x − a ) with k b (cid:29) k f . Finally, substrate binding will also af-fect the fluctuations of the orientational modes of thedumbbell through the coefficients V α . Such an effect wasfor instance suggested for enzymes like urease, which isknown to have a ‘flap’ that is closed when the enzyme isbound and open otherwise [41].For concreteness, we now estimate the contribution to A by the different specific structural modifications of theenzyme discussed above. We first focus on the compres-sional modes of fluctuations and on the associated termsin Eq. (4). When the change in the diffusion coefficientoriginates from a reduction in the equilibrium distancebetween the subunits, we can expand the expression for A in the limit of small relative deformation ( δx (cid:28) a )and large potential stiffness ( k f (cid:29) k B T /a ) [31] to find A c = G · δx/a with the dimensionless parameter definedas G = a (cid:34) − m (cid:48) + 16 γ w (cid:18) ln γ w (cid:19) (cid:48) (cid:35) (cid:20) m − γ w (cid:21) − , (15)where the mobility tensors are evaluated at the equi-librium length x = a (i.e. the configuration averaginghas been implemented). While G is a purely geometricquantity that is typically of order unity, the result showsthat the extent of the average deformation directly con-trols the magnitude of A . For the case when the sub-strate binding makes the protein stiffer, Fig. 3 showsa plot of the relevant factor A s as a function of the ra-tio between the two stiffnesses k b /k f ; we observe thatthis factor is typically of order one and that it increaseswhen the two stiffnesses deviate from one another sub-stantially. We can also find a closed form expression inthe limit of very large k f and k b with a finite difference δk as A s = H · [ k B T / ( k f a )] · ( δk/k f ) where H is a di-mensionless coefficient of order unity [31]. Finally, if theorientational fluctuations of the dumbbell are affected bysubstrate binding, we can deduce the contribution to therelative change of the diffusion coefficient in the simplecase where the subunits are freely fluctuating in the freestate ( V α = 0) and constrained otherwise ( V α > A r (cid:39) k B Tk f a (cid:88) α =1 , V α, f ( V α, b − V α, f ) J α , (16)where J α are dimensionless coefficients of order unity[31]. Conclusion.—
We have proposed a simple model tostudy the effect of asymmetry on the fluctuation-inducedhydrodynamic coupling between the different parts of amodel enzyme. We consider the interplay between its in-ternal and external degrees of freedom and calculate thecorrections to the overall diffusion coefficient that origi-nate from the compressional and orientational degrees offreedom, and that are controlled by the structural asym-metries of the molecule. This generic model can be usedto describe the mechanochemical cycles explored by en-zyme molecules when placed in the presence of substratemolecules. We show how substrate binding and unbind-ing can lead to diffusion enhancement, and confront ourtheoretical predictions to recently published experimen-tal measurements. Our minimal model, that contains allthe required physical ingredients, completes the existingtheoretical picture that failed to explain consistently theexperimental observations so far.P. I. and R.G. acknowledge financial support from theUS National Science Foundation under MRSEC GrantNo. DMR-1420620. We benefited from fruitful discus-sions with A. Sen and K. K. Dey. [1] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts,and P. Walter,
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Supplementary Information
Pierre Illien, , Tunrayo Adeleke-Larodo, Ramin Golestanian, Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK Department of Chemistry, The Pennsylvania State University, University Park, PA 16802, USA
COMBINATION OF MOBILITY TENSORS USED IN THE MAIN TEXT M (ˆ u , ˆ u , ˆ n ) = M + M + 2 M (cid:39) [ m + m ( ˆ n · ˆ u ) + m ( ˆ n · ˆ u ) + m (ˆ u · ˆ u ) + . . . ] , (S1) W (ˆ u , ˆ u , ˆ n ) = M + M − M (cid:39) [ w + w ( ˆ n · ˆ u ) + w ( ˆ n · ˆ u ) + w (ˆ u · ˆ u ) + . . . ] , (S2) Γ (ˆ u , ˆ u , ˆ n ) = M − M (cid:39) [ γ + γ ( ˆ n · ˆ u ) + γ ( ˆ n · ˆ u ) + γ (ˆ u · ˆ u ) + . . . ] , (S3) Ψ (1) (ˆ u , ˆ u , ˆ n ) = M (cid:39) [ ψ (1)0 + ψ (1)1 ( ˆ n · ˆ u ) + ψ (1)2 ( ˆ n · ˆ u ) + ψ (1)12 (ˆ u · ˆ u ) + . . . ] , (S4) Ψ (2) (ˆ u , ˆ u , ˆ n ) = M (cid:39) [ ψ (2)0 + ψ (2)1 ( ˆ n · ˆ u ) + ψ (2)2 ( ˆ n · ˆ u ) + ψ (2)12 (ˆ u · ˆ u ) + . . . ] . (S5)In the present work, we will only consider the leading order terms of these expansions, and will consider the correctionsin a later paper.For clarity, we can specify the functions m , γ and w in the Oseen limit, which corresponds to thesituation where the two subunits are spherical with respective radii a and a , in the limit of x (cid:29) a , and neglectingthe terms involving the orientation ˆ n : m (cid:39) πη (cid:18) a + 1 a (cid:19) + 14 πηx (S6) γ (cid:39) πη (cid:18) a − a (cid:19) (S7) w (cid:39) πη (cid:18) a + 1 a (cid:19) − πηx (S8) AVERAGE OF EQ. (3)
Starting from Eq. (3) of the main text, we use the centre of mass and elongation coordinates, R = ( x + x ) / x = x − x , to find the Smoluchowski equation for P ∂ t P = k B T ∇ R · M · ∇ R P + 12 ∇ R · Γ · ( ∇ x U ) P + k B T ∇ R · Γ · ∇ x P + ∇ x · Γ · ∇ R P )+ ∇ x · W · [ k B T ∇ x P + ( ∇ x U ) P ] + (cid:88) α,β =1 , R α · M αβ RR · [ k B T R β P + ( R β U ) P ]+ (cid:88) α =1 , (cid:110) ∇ α · M αβ TR · [( R β U ) P + k B T R β P ] + R α · M αβ RT · [( ∇ β U ) P + k B T ∇ β P ] (cid:111) . (S9)Within the expansion used in Eqs. (S1) to (S5), the terms involving the tensors M αβ TR and M αβ RT will not contributeat leading order in the moment expansion that follows. We will consider these contributions in a later publication.Using the generic form for the interaction potential U = v ( x ) + v ( x ) ˆ n · ˆ u + v ( x ) ˆ n · ˆ u + v ( x )ˆ u · ˆ u , uponaveraging Eq. (S9) over x we find ∂ t P = k B T ∇ R · (cid:104) M (cid:105) · ∇ R P + 12 ∇ R · (cid:34)(cid:28) Γ v ( x ) x (cid:29) · ( − ˆ n ˆ n ) · ˆ u P + (cid:28) Γ v ( x ) x (cid:29) · ( − ˆ n ˆ n ) · ˆ u P (cid:35) − k B T ∂ R i (cid:15) jkl (cid:20) R l (cid:18)(cid:28) Γ ij x (cid:29) ˆ n k P (cid:19) − (cid:28) R l (cid:18) Γ ij x ˆ n k (cid:19)(cid:29) P (cid:21) − k B T n × R ) · ( − ˆ n ˆ n ) · (cid:28) Γ x (cid:29) · ∇ R P + k B T ( ˆ n × R ) i ( − ˆ n ˆ n ) ij (cid:15) klm (cid:20) R m (cid:18)(cid:28) W jk x (cid:29) ˆ n l P (cid:19) − (cid:28) R m (cid:18) W jk x ˆ n l (cid:19)(cid:29) P (cid:21) − ( ˆ n × R ) · ( − ˆ n ˆ n ) · (cid:88) α =1 , (cid:28) W x v α ( x ) (cid:29) · ( − ˆ n ˆ n ) · ˆ u α P + R · [ (cid:104) Ψ (1) v (cid:105) · (ˆ u × ˆ u ) P + (cid:104) Ψ (1) v (cid:105) · (ˆ u × ˆ n ) P − (cid:104) M v (cid:105) · (ˆ u × ˆ u ) P + (cid:104) M v (cid:105) · (ˆ u × ˆ n ) P ]+ R · [ −(cid:104) Ψ (2) v (cid:105) · (ˆ u × ˆ u ) P + (cid:104) Ψ (2) v (cid:105) · (ˆ u × ˆ n ) P + (cid:104) M v (cid:105) · (ˆ u × ˆ u ) P + (cid:104) M v (cid:105) · (ˆ u × ˆ n ) P ] , (S10)where P = (cid:82) d x x P and (cid:104)·(cid:105) = P (cid:82) d x x · P . We define the lowest order moments of the average distribution withrespect to the three unit vectors ˆ n , ˆ u and ˆ u as ρ ≡ (cid:82) ˆ n , ˆ u , ˆ u P , p ≡ (cid:82) ˆ n , ˆ u , ˆ u ˆ n P and p α ≡ (cid:82) ˆ n , ˆ u , ˆ u ˆ u α P and obtainthe respective evolution equations in the moment expansion of (S10) ∂ t ρ = k B T (cid:104) m (cid:105) ∇ R ρ + k B T (cid:68) γ x (cid:69) ∇ R · p + 13 (cid:20)(cid:28) γ v x (cid:29) ∇ R · p + (cid:28) γ v x (cid:29) ∇ R · p (cid:21) , (S11) ∂ t p i = − k B T (cid:68) γ x (cid:69) ∂ R i ρ − k B T (cid:68) w x (cid:69) p i − (cid:88) α =1 , (cid:28) w v α x (cid:29) p αi , (S12) ∂ t p αi = − k B T (cid:104) ψ ( α )0 (cid:105) p αi + 19 (cid:28) γ v α x (cid:29) ∂ R i ρ −
23 [ (cid:104) ψ ( α )0 v (cid:105) p βi + (cid:104) ψ ( α )0 v α (cid:105) p i − (cid:104) M αβ RR 0 v (cid:105) p βi ] . (S13)where we have used the relation (cid:104) U (cid:48) φ ( x ) (cid:105) = k B T (cid:28) φ (cid:48) ( x ) + 2 φ ( x ) x (cid:29) , (S14)valid for any function φ of the radial coordinate and under the assumption that the x -dependence of P is Boltzmann-like i.e. P ∝ e − U/k B T . The resulting hierarchy of equations is closed with the following prescription for the secondmoments (cid:90) ˆ n , ˆ u , ˆ u P n i n j (cid:39) δ ij ; (cid:90) ˆ n , ˆ u , ˆ u P u αi u βj (cid:39) δ αβ δ ij ; (cid:90) ˆ n , ˆ u , ˆ u P u αi n j (cid:39) , (S15)and similarly for higher order moments. We neglected any spatial derivatives of the polarisation fields in Eqs. (S12)and (S13). A closed equation for the density ρ can be obtained by taking the stationary limits of these equations, andwe get ∂ t ρ ( R ; t ) = D eff ∇ R ρ . Using the specific form of the potential U = k ( x − a ) [1+ (cid:80) α =1 , V α ˆ n · ˆ u α + V ˆ u · ˆ u ],we find the expression of D eff given in the main text [Eq. (4)]. DEFINITION OF K α K α = 19 (cid:40) (cid:104) γ ϕ/x (cid:105) (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105)(cid:104) ψ ( α )0 (cid:105) + (cid:104) γ ϕ/x (cid:105)(cid:104) γ /x (cid:105) (cid:104) ψ ( α )0 ϕ (cid:105)(cid:104) ψ ( α )0 (cid:105) − (cid:104) γ ϕ/x (cid:105)(cid:104) γ /x (cid:105) (cid:104) w ϕ/x (cid:105)(cid:104) ψ ( α )0 (cid:105) − (cid:104) w ϕ/x (cid:105)(cid:104) w /x (cid:105) (cid:104) ψ ( α )0 ϕ (cid:105)(cid:104) ψ ( α )0 (cid:105) (cid:41) (S16)where we define ϕ ( x ) = [ xa − .0 RELATION BETWEEN THE PRE-AVERAGED FRICTION TENSORS TO THE PRE-AVERAGEDMOBILITY TENSORS
We start from the following relation between the pre-averaged friction tensors and the pre-averaged mobility tensors (cid:18)(cid:10) z (cid:11) (cid:10) z (cid:11)(cid:10) z (cid:11) (cid:10) z (cid:11)(cid:19) = (cid:18)(cid:10) m (cid:11) (cid:10) m (cid:11)(cid:10) m (cid:11) (cid:10) m (cid:11)(cid:19) − , (S17)and, using the definitions of M , W and Γ on the one hand, and Z , Y , and ζ on the other hand (see main text), wefind (cid:104) y (cid:105) = 4 (cid:104) m (cid:105)(cid:104) w (cid:105) (cid:104) m (cid:105) − (cid:104) γ (cid:105) ; (cid:104) z (cid:105) = 4 (cid:104) w (cid:105)(cid:104) w (cid:105) (cid:104) m (cid:105) − (cid:104) γ (cid:105) ; (cid:104) ζ (cid:105) = 4 (cid:104) Γ (cid:105)(cid:104) w (cid:105) (cid:104) m (cid:105) − (cid:104) γ (cid:105) . (S18) BINDING OF THE ENZYME TO SUBSTRATE AND PRODUCT MOLECULES
The simplified mechanochemical cycle we present in the text is the following:E + S α ( S ) −−− (cid:42)(cid:41) −−− β C γ −−− (cid:42)(cid:41) −−− δ ( P ) E + P , (S19)where C represents the enzyme either bound to a substrate or a product molecule. We first determined the concen-trations of S and P in the equilibrium state, in which the following conditions are satisfied: α ( S ) p f = β (1 − p f ) (S20) γ (1 − p f ) = δ ( P ) p f , (S21)where p f is the probability to find the enzyme in its free state. α ( S ) increases linearly with S as α ( S ) = α S , and δ ( P )increases linearly with P as δ ( P ) = δ P , so that the solution of the system gives the expression of S/P at equilibrium: SP = δ βα γ . (S22)If we denote by U S (resp. U P ) the conformational energy of the enzyme when it is bound to a substrate (resp.product) molecule, we write the probability of a given conformation C under the form p ( C ) ∝ e − U f /k B T + SK S , e − U S /k B T + PK P , e − U P /k B T , (S23)where we defined K S , = β/α and K P , = γ/δ . For any function of a conformational state Φ( C ), its average writes (cid:104) Φ (cid:105) = (cid:82) C Φ e − U f /k B T + SK S , (cid:82) C Φ e − U S /k B T + PK P , (cid:82) C Φ e − U P /k B T Z f + SK S , Z S + PK P , Z P , (S24)with Z X = (cid:82) C e − βU X . We then deduce (cid:104) Φ (cid:105) = (cid:104) Φ (cid:105) f + SS + K S K P P + K S [ (cid:104) Φ (cid:105) S − (cid:104) Φ (cid:105) f ] + PP + K P K S S + K P [ (cid:104) Φ (cid:105) P − (cid:104) Φ (cid:105) f ] , (S25)where we define K S = K S , (cid:82) C e − βU f (cid:82) C e − βU S ; K P = K P , (cid:82) C e − βU f (cid:82) C e − βU P . (S26)This result can be written under a simpler form if we assume that U P = U S = U b : (cid:104) Φ (cid:105) = (cid:104) Φ (cid:105) f + (cid:32) SS + K S K P P + K S + PP + K P K S S + K P (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) ≡ f ( S,P ) [ (cid:104) Φ (cid:105) b − (cid:104) Φ (cid:105) f ] , (S27)which defines the function f ( S, P ) introduced in the main text. We finally consider two limits of this result:1 • at short times, very few product molecules have been formed ( P (cid:39) S (cid:39) S ), and we find (cid:104) Φ (cid:105) = (cid:104) Φ (cid:105) f + S S + K S [ (cid:104) Φ (cid:105) b − (cid:104) Φ (cid:105) f ] , (S28)which corresponds to the expression given in the main text, where we neglected the formation of productmolecules. • at long times, when chemical equilibrium is reached, the concentrations of S and P are given by S = K S K S + K P S ; P = K P K S + K P S , (S29)and we find (cid:104) Φ (cid:105) = (cid:104) Φ (cid:105) f + S S + ( K S + K P ) [ (cid:104) Φ (cid:105) b − (cid:104) Φ (cid:105) f ] , (S30)so that the effective equilibrium constant used in the main text reads K = 12 ( K S + K P ) = 12 (cid:34) K S , (cid:82) C e − βU f (cid:82) C e − βU S + K P , (cid:82) C e − βU f (cid:82) C e − βU P (cid:35) . (S31) CONTRIBUTIONS TO THE DIMENSIONLESS COEFFICIENT A Change in the diffusion coefficient
Here, we estimate the change in the diffusion coefficient of the dumbbell due to the structural modifications inducedby substrate binding presented above. Neglecting the effect of the orientational degrees of freedom in Eq. (4) fromthe main text, we write the effective diffusion coefficient of the enzyme for an arbitrary substrate concentration as D ( S ) = k B T (cid:104) m (cid:105) − k B T (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) , (S32)where the averages are to be understood with the weight defined in Eq. (S27). The change in the diffusion coefficientis given as ∆ D = D ( S ) − D ( S = 0) = (cid:34) k B T (cid:104) m (cid:105) − k B T (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) (cid:35) − (cid:34) k B T (cid:104) m (cid:105) f − k B T (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) f (cid:35) , (S33)where indexed brackets denote an average with the probability distribution corresponding to the free state as definedabove. Using Eq. (S30) to write the average quantities in terms of the corresponding pure averages in the free andbound states in the long-time limit, we find ∆ DD = A · S S + K , (S34)where A = (cid:104) (cid:104) m (cid:105) f (cid:16) (cid:104) m (cid:105) b −(cid:104) m (cid:105) f (cid:104) m (cid:105) f (cid:17)(cid:105) − (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) f (cid:20) (cid:16) (cid:104) γ /x (cid:105) b −(cid:104) γ /x (cid:105) f (cid:104) γ /x (cid:105) f (cid:17) − (cid:18) (cid:104) w /x (cid:105) b − (cid:104) w /x (cid:105) f (cid:104) w /x (cid:105) f (cid:19)(cid:21)(cid:104) (cid:104) m (cid:105) f − (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) f (cid:105) , (S35)to the lowest order in difference between free and bound averages. We now consider the different effects separately.2 Modification of the equilibrium distance between the subunits
Substrate binding is expected to reduce the equilibrium distance between the subunits that constitute the dumbbell.We first estimate the contribution of this effect to the diffusion coefficient of the dumbbell, by using the explicit(harmonic) form for the potentials U f and U b : U f = 12 k f ( x − a ) ; U b = 12 k f [ x − ( a − δx )] , (S36)and expand the expression of ∆ D/D [Eq. (S34)] in the small deformation limit ( δx (cid:28) a ), in which the followingexpression (cid:104) Φ (cid:105) b − (cid:104) Φ (cid:105) f = k f δxk B T ( (cid:104) x (cid:105) f (cid:104) Φ (cid:105) f − (cid:104) x Φ (cid:105) f ) + O [( δx ) ] , (S37)is to be incorporated for each average quantity. At linear order in δx , we find the change in the diffusion coefficientof the enzyme due to its compression A c (cid:39) (cid:2) (cid:104) m (cid:105) f ( (cid:104) x (cid:105) f (cid:104) m (cid:105) f − (cid:104) xm (cid:105) f ) (cid:3) − (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) f (cid:2) (cid:104) x (cid:105) f (cid:104) γ /x (cid:105) f − (cid:104) γ (cid:105) f ) − (cid:0) (cid:104) x (cid:105) f (cid:10) w /x (cid:11) f − (cid:104) w /x (cid:105) f (cid:1)(cid:3)(cid:104) (cid:104) m (cid:105) f − (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) f (cid:105) k f δxk B T . (S38)In the limit where the stiffness of the spring linking the two subunits of the enzyme is very large ( k f (cid:29) k B T /a ), wecan expand the averages (cid:104)·(cid:105) f using Laplace’s method and rewrite the amplitude of the relative increase of the diffusioncoefficient as A c = G · δxa , (S39)where G = a (cid:34) − m (cid:48) + 16 γ w (cid:18) ln γ w (cid:19) (cid:48) (cid:35) (cid:20) m − γ w (cid:21) − , (S40)and the mobility coefficients and their derivatives are evaluated at the equilibrium position x = a . Modification of the potential stiffness
Using the explicit expressions of the potentials U f and U b U f = 12 k f ( x − a ) ; U b = 12 k b ( x − a ) , (S41)in Eq. (S34), we can calculate the contribution due to change in stiffness.This contribution can first be estimated numerically, by using different approximations for the mobility coefficientsof the subunits of the dumbbell, when they are assumed to be spherical. In the far-field (Oseen) limit, these mobilitycoefficients are given by [29] M αα TT = 16 πηa α ; M αβ TT = 18 πηx ( + ˆ n ˆ n ) ( α (cid:54) = β ) , (S42)where a α is the radius of subunit α . Defining M αβ TT = M αβ I + M αβ D ˆ n ˆ n , the corrections to these leading order terms3can be obtained as a series expansion in powers of 1 /x [39]: M αα I = 16 πηa α − a β πηx − a β (10 a α − a α a β + 9 a β ) πηx − a β (35 a α − a α a β + 6 a β ) πηx , (S43) M αα D = − a β πηx + 532 (8 a α − a β ) a β πηx −
148 (20 a α − a α a β + 9 a β ) a β πηx −
196 (175 a α + 1500 a α a β − a α a β + 18 a β ) a β πηx , (S44) M αβ I = 18 πηx + 124 a α + a β πηx + 7768 (80 a α − a α a β + 80 a β ) a α a β πηx , (S45) M αβ D = 18 πηx − a α + a β πηx + 258 a α a β πηx −
58 ( a α + a β ) a α a β πηx + 1768 (400 a α − a α a β + 400 a β ) a α a β πηx . (S46)In the Smoluchowski description of the stochastic dynamics of the system, and for the particular case of sphericalsubunits, we can refine the approximate forms of the mobility tensors and use A = A I + A D ˆ n ˆ n instead of A (cid:39) a for A = M , W and Γ . We will consider this approximation in greater details in a later publication. We can showthat Eq. (S32) still holds, with m = M I + M D / γ = Γ I and w = W I . These expressions for the mobility functionsare used to produce the plots shown in the main text.We can also estimate analytically the relative change of the diffusion coefficient due to an increase in the potentialstiffness in the limit of k f → ∞ and k b → ∞ with a fixed difference δk = k b − k f : A s = H · ε · δkk f , (S47)where ε = (cid:112) k B T / ( k f a ) is a dimensionless number that characterises the amplitude of the thermal fluctuations ofthe dumbbell elongation around its equilibrium value, and where the dimensionless coefficient H reads H = − (cid:20) a m (cid:48)(cid:48) am (cid:48) + 13 γ w − a γ γ (cid:48) w − a γ γ (cid:48)(cid:48) w + a w (cid:48)(cid:48) γ w (cid:21) (cid:20) m − γ w (cid:21) − , (S48)with the mobility coefficients and derivatives evaluated at the equilibrium position x = a . Hindering of orientational fluctuations
We finally estimate the contribution to A coming from changes in orientational fluctuations through the dimension-less coefficients V α . Denoting by V α, f (resp. V α, b ) the value of the coefficients V α in the free (resp. bound) state, wefind at leading order in the corrections [ (cid:104) Φ (cid:105) b − (cid:104) Φ (cid:105) f ] the contribution to the dimensionless coefficient A that originatesfrom constraints on the orientational fluctuations of the subunits: A r = k B T (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) f (cid:80) α =1 , (cid:16) k f a k B T (cid:17) V α, f ( V α, b − V α, f ) K αk B T (cid:104) m (cid:105) f − k B T (cid:104) γ /x (cid:105) (cid:104) w /x (cid:105) f (cid:20) − (cid:80) α =1 , (cid:16) k f a k B T V α (cid:17) K α (cid:21) (S49)As for the coefficients A c and A s , this expression can be estimated in the limit where k f (cid:29) k B T /a : A r (cid:39) k B Tk f a (cid:88) α =1 , V α, f ( V α, b − V α, f ) J α (S50)with J α = γ w m − γ w ( a ψ ( α )0 + w )( γ (cid:48)(cid:48) w a − γ w (cid:48)(cid:48) a + 2 γ (cid:48) w ) aγ w ψ ( α )0 (S51)with the mobility coefficients and derivatives evaluated at the equilibrium position x = aa