How kinesin waits for ATP affects the nucleotide and load dependence of the stepping kinetics
Ryota Takaki, Mauro L. Mugnai, Yonathan Goldtzvik, D. Thirumalai
HHow kinesin waits for ATP affects the nucleotideand load dependence of the stepping kinetics
Ryota Takaki a , Mauro L. Mugnai b , Yonathan Goldtzvik b , and D. Thirumalai b a Department of Physics, The University of Texas at Austin, Austin, TX, USA; b Department of Chemistry, The University of Texas at Austin, Austin, TX, USA
Conventional Kinesin (Kin-1), which is responsible for directionaltransport of cellular vesicles, takes multiple nearly uniform 8.2 nmsteps by consuming one ATP molecule per step as it walks towardsthe plus end of the microtubule (MT). Despite decades of intensive ex-perimental and theoretical studies there are gaps in the elucidationof key steps in the catalytic cycle of kinesin. For example, how themotor waits for ATP to bind to the leading head has become contro-versial. Two experiments using a similar protocol, which follow themovement of a large gold nanoparticle attached to one of the motorheads, have arrived at different conclusions. One of them (1) assertsthat kinesin waits for ATP in a state with both heads bound to theMT, whereas the other (2) shows that ATP binds to the leading headafter the trailing head is detached. In order to discriminate betweenthese two scenarios, we developed a minimal model, which analyti-cally predicts the outcomes of a number of experimental observablesquantities, such as the distribution of run length [ P ( n ) ], the distribu-tion of velocity [ P ( v ) ], and the randomness parameter as a functionof an external resistive force ( F ) and ATP concentration ([T]). We findthat P ( n ) is insensitive to the waiting state of kinesin. The bimodalvelocity distribution P ( v ) depends on the ATP waiting states of ki-nesin. The differences in P ( v ) as a function of F between the twomodels may be amenable to experimental testing. Most importantly,we predict that the F and [T] dependence of the randomness param-eters differ qualitatively depending on whether ATP waits with bothheads bound to the MT or with detached tethered head. The ran-domness parameters as a function of F and [T] can be quantitativelymeasured from stepping trajectories with very little prejudice in dataanalysis. Therefore, an accurate measurement of the randomnessparameter and the velocity distribution as a function of load and nu-cleotide concentration could resolve the apparent controversy, thusproviding insights into the waiting state of kinesin for ATP. kinesin | molecular motors | chemomechanical coupling | randomnessparameter K inesin-1 (Kin1) is an archetypal cellular transporter,which moves along the microtubule (MT) to shuttle cargotowards the cellular periphery. In the last quarter of century,a number of spectacular experimental studies (3–7) have re-vealed many of the salient features of Kin1 structure andmotility. (i) Kin1 is a homodimer made up of two ATPase andMT-binding heads. A key structural element, the neck-linker(NL) undergoes an order/disorder transition during the cat-alytic cycle termed “NL docking". The distal tail forms a coiledcoil which is responsible for dimerization and is also involvedin cargo binding (8). (ii) Remarkably, the motor takes almostprecisely 8.2 nm steps (7), which is commensurate with thespacing between two adjacent αβ dimers – the building blocksof the MT filament. (iii) For each diffusional encounter withthe MT, Kin1 takes multiple steps before detaching, a featuretermed processivity (9). (iv) In the absence of resistive load( F ), Kin1 moves nearly unidirectionally (backward steps are rare) towards the plus end of the MT (10), and predominantlyalong a single protofilament (11). In addition, the velocity( v ) distribution is roughly Gaussian with a peak typically inthe range (100 - 1000) nm · s − depending on ATP concen-tration (12); the mean velocity is much larger than what isfound in other motors such as Myosin V and Dynein. Asthe resisting load increases, the probability that the motortakes backward steps becomes more prominent, reaching 0 . F S ≈ PNAS |
August 22, 2019 | vol. XXX | no. XX | a r X i v : . [ q - b i o . S C ] A ug inds. However, in order to discriminate between the 1HBand 2HB ATP waiting states, it is necessary to monitor thelocation of the tethered head at the time of ATP binding,which requires experiments with high temporal and spatialresolution.The development of an experimental technique in whicha large gold nanoparticle, AuNP, (between (20 - 40) nm indiameter) is attached to one of the heads, has made it possibleto track indirectly the position of the tethered head duringthe stepping process as a function of ATP concentration. Bytracking the location of the AuNP, either via interferometricscattering microscopy (iSCAT) (1) or total internal-reflectiondark-field microscopy (2), two groups have achieved a degree oftemporal and spatial resolution necessary to resolve the waitingstate of kinesin. From the analysis of the AuNP movementat different ATP concentrations, Micolajczky et al. arguedthat the motor waits in the 2HB state when the concentrationof ATP is ≥ µ M. The 2HB → et al. established thatATP binds to the LH only after the TH detaches from theMT. In other words, Kin1 waits for ATP in the 1HB state.In addition, computer simulations, using coarse-grained (CG)models for motors in general (32, 33) and kinesin in particularhave provided insights into their functions. In particular, CGmodels that accurately reproduce several features found inexperiments (16, 17, 34–36), have shown that the TH doesspontaneously detach but does not walk towards the plus end ofthe MT until neck linker docks to the LH, which requires ATPbinding to the leading head. These findings support the 1HBconformation as Kin1 ATP waiting state. The contradictoryfindings reported in (1, 2) and alluded to by Sindelar (37),leave the vexing question posed in the previous paragraphunanswered. This basic question needs to be fully answeredin order to achieve a complete understanding of the steppingmechanism of conventional kinesin.It is unclear whether the differing conclusions reached in therecent experimental studies (1, 2) arise because of the discrep-ancies in the constructs of the kinesin, the method of analysisof the trajectories, or due to the variations in the temporalresolution achieved in the experiments. Isojima et al. used acys-lite motor in order to control the location of the linkage be-tween the motor and the AuNP. In contrast, Mickolajczyk et al. used a WT Kin1, whose N-terminus was extended with an Avitag which is linked to the AuNP through a streptavidin-biotincomplex. Moreover, because of the higher temporal resolutionin the dark field microscopy experiments (2), Isojima et al. could discern the 1HB state by simultaneously monitoringthe transverse fluctuations directly from the trajectories in astraightforward manner. On the other hand, Mickolajczyk etal. (1) relied on Hidden Markov Models (HMMs) to extractinformation from the stepping trajectories.In order to discriminate between the contrasting interpreta-tions of these experiments, it is desirable to consider quantitiesthat are straightforward to measure, and that do not requireindirect techniques of data analysis. Ideally, a theoreticalstudy capable of describing both scenarios (1HB and 2HBwaiting state for ATP) should be able to identify which ob-servable might be used to discriminate between the proposed cycles for kinesin. Here, we use a simple and accurate modelfor kinesin stepping and calculate analytically a number ofstandard measurable quantities, such as the run length ( n )distribution, P ( n ), velocity ( v ) distribution, P ( v ), and therandomness parameter as a function of ATP concentration(denoted as [T] from now on) and resistive force F . We showthat P ( n ) is independent of [T], and P ( v ) as a function of[T] and F are qualitatively similar for both the 1HB and2HB models but differ quantitatively, a discrepancy that isamenable to experimental test. Remarkably, we predict thatthe mechanical and chemical randomness parameters, whichare defined from readily measurable quantities, could be usedto discriminate between the two scenarios. In particular, wefind that both the mechanical and chemical randomness pa-rameters at different [T] and F are qualitatively different forthese two scenarios in which Kin1 waits for ATP either in the1HB or in the 2HB state. Thus, we propose that measurementsof the randomness parameters and P ( v ) as a function of [T]and F should unambiguously allow one to distinguish betweenthe two very different ATP waiting states of Kin1. Results
We begin by presenting some nomenclature. We refer to thescenario in which ATP binds to the LH of kinesin when bothheads are attached to the MT as the “2HB model”, whereas the“1HB model” refers to the alternative sequence of events, inwhich the detachment of the TH of kinesin precedes the bindingof ATP to the LH. In order to calculate P ( v ) and P ( n ) wecreated two versions of what is perhaps the simplest chemicalkinetics model for a molecular motor [Fig. 1(c) and (d)], one forthe 2HB and one for the 1HB model. The difference betweenthe two lies in the the dependence on ATP concentration ofthe kinetic rates. In the 2HB model the transition to the1HB state occurs only after ATP binds to the leading head[Fig. 1(c)], therefore, the 2HB → k + and k − , as well as γ are assumed to depend on [T] [Fig. 1(d)].We use Michaelis-Menten kinetics to describe ATP bindingand account for the effect of external load on the rates byadopting the Bell model. In order to distinguish betweenthe parallel component of the vectorial load applied to themotor, which introduce the symbols k and ⊥ , respectively [seeFig. 1(b)]. For the 2HB model, k { [T] } = k [T] K T +[T] , k + ( F ) = k +0 e − βFd + , k − ( F ) = k − e βFd − , and γ ( F ) = γ e | F | /F d , where d ± = d ±k F k /F and the load F d = ( | F | k B T ) / ( F ⊥ d γ ). In thecase of the 1HB model [Fig. 1(d)], k is a constant, independentof [T] and load, k + = k +0 [T] K T +[T] e − βFd + , k − = k − [T] K T +[T] e βFd − ,and γ = γ [T] K T +[T] e F/F d . Note that in both the scenarios wehave assumed that the 2HB → P ( v ) and P ( n ) weobtain the stationary fluxes for forward stepping, backwardstepping, and detachment. The motor is viewed as a randomwalker starting in the 2HB state at the MT site i . A steady-state probability distribution of occupying the 2HB and 1HBstates is enforced by replenishing the 2HB state of all thewalkers that step forward or backward (reaching i + 1 and | Takaki et al. a )( b ) k + { F } k { F } { F } i i+1 i-1 k { [T] } ( c ) i i+1 i-1 k k + { [T] , F } k { [T] , F } { [T] , F } OUT ~F ? ~F k Bead
OUT ( d ) ~F + . . - Fig. 1. (a) Schematic representation of a kinesin motor walking hand over hand on the microtubule (MT). The tethered head detaches, undergoes diffusion, and passes theleading head (LH), and reattached to the target binding site that is roughly 16.4 nm from the starting position, resulting in a net displacement of 8.2 nm step. In the process oneATP molecule is hydrolyzed. (b) Decomposition of the resistive force applied to the bead attached to the coiled coil of kinesin into perpendicular ( ⊥ ) and parallel ( k ) direction tothe MT. (c) Kinetic scheme describing the ATP-waiting state showing that binding to the LH occurs when both the heads are bound to the MT in the 2HB state. (d) Same as (c)except ATP binds when kinesin is in 1HB state. In both the scenarios the i-1 and i+1 state are equivalent to i in that they correspond to both the heads bound to the track. Thedifference is in the state that waits for ATP. The state labeled OUT represents an absorbing state. i −
1, respectively) or detach (38, 39), dP dt = − kP + ( γ + k + + k − ) P = 0 dP dt = − ( γ + k + + k − ) P + kP = 0 , [1]The normalization condition implies that P + P =1. The solution of Eq.(1) gives P = kk + k + + k − + γ . Thestationary fluxes for forward stepping ( J + ), backward stepping( J − ), and detachment ( J γ ) are computed by multiplying thesteady-state probability of being in state 1HB ( P ) times k + , k − , and γ (38, 39), J ± = kk T k ± , J γ = kk T γ, [2]where k T = k + k + + k − + γ .The average velocity and run length are given by V = s ( J + − J − ) and L = V /J γ , respectively, where s = 8 . V = V max [T] K M + [T] , L = ( k + − k − ) sγ , [3]for both the 2HB and 1HB model. The maximum ve-locity at saturating ATP concentration for the 2HB and1HB model are given by V max = k ( k + − k − ) sk + k + + k − + γ and V max =( k +0 e − βF + d − k − e βF − d ) ks/k T , respectively. The concentra-tions at which the velocity of Kin1 is half-maximal are given by K M = K T ( k + + k − + γ ) k + k + + k − + γ for the 2HB model [Fig.1(c)] and K M = K T for 1HB model [Fig.1(d)]. Run length distribution, P ( n ) . In order to solve for the runlength and velocity distributions, we construct the joint prob-ability [ P ( m, l )] that the motor takes m forward steps and l backward steps before detachment (see the SupplementaryInformation (SI) for details), P ( m, l ) = ( m + l )! m ! l ! (cid:16) J + J T (cid:17) m (cid:16) J − J T (cid:17) l (cid:16) J γ J T (cid:17) . [4]In the above equation, J + /J T ( J − /J T ) is the probability oftaking a forward (backward) step starting from the 2HB state,and J T = J + + J − + J γ . Similarly, J γ /J T is the probabilitythat a motor in the 2HB state detaches. The number ofall the possible ways in which a sequence of m forward and l backward steps can be realized is accounted for by the binomialfactor. If the run length is n = m − l , then P ( n ) is given by P ( n ) = P ∞ m,l =0 P ( m, l ) δ m − l,n , where δ m − l,n is the Kroneckerdelta function. By carrying out the summation we obtain, P ( n ≷
0) = (cid:16) J ± J T + p J T − J + J − (cid:17) | n | J γ p J T − J + J − . [5]Note that the functional form of P ( n ≷
0) is independentof the model considered – it is the dependence of the fluxeson [T] and F that separates the 2HB and 1HB model. Wenote that the expression for P ( n ) obtained here is equivalentto the one obtained previously(15, 40), which can be derived Takaki et al.
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August 22, 2019 | vol. XXX | no. XX | y substituting the rate of forward step, backward step, anddetachment to the corresponding fluxes defined in Eq.(2). Velocity distribution, P ( v ) . In order to calculate P ( v ), wefirst compute f ( m, l, t ), which is the joint probability densityfor detaching during a time interval from t to t + dt after themotor takes m forward and l backward steps. Let f + ( t ) bethe probability density of taking a forward step between t and t + dt , given that at t = 0 the motor is in the 2HB state.Similarly, the probability density for stepping backward andfor detachment are denoted by f − ( t ) and f γ ( t ). We show in theSI that f + ( t ), f − ( t ), and f γ ( t ) are linear combinations of twoexponential functions with rates ξ = k and ξ = k + + k − + γ (see Fig. 1 for the definition of the rates). The probabilitydensity f ( m, l, t ) is given by, f ( m, l, t ) = ( m + l )! m ! l ! Z t dt m + l Z t m + l dt m + l − · · · Z t dt Z t dt m Y i =1 f + ( t i − t i − ) m + l Y i = m +1 f − ( t i − t i − ) f γ ( t − t m + l ) . [6]As detailed in the SI, the solution of the integral equation inEq.(6) is, f ( m, l, t ) = γ √ πm ! l ! e − ξ ξ t t m + l k m + l +1 ( k + ) m ( k − ) l | ξ − ξ | m + l +1 p | ξ − ξ | tI m + l + (cid:16) | ξ − ξ | t (cid:17) , [7]where I m + l + (cid:16) | ξ − ξ | t (cid:17) is the modified Bessel function ofthe first kind. The velocity distribution may be obtained bychanging the variables to v = ( m − l ) /t , which gives, P ( v >
0) = ∞ X m,lm>l m − lv γ √ πm ! l ! e − ξ ξ m − lv (cid:0) m − lv (cid:1) m + l + k m + l +1 ( k + ) m ( k − ) l | ξ − ξ | m + l + I m + l + (cid:0) | ξ − ξ | m − lv (cid:1) . [8]The expression for P ( v <
0) is presented in the SI. Note thatboth Eq. (7) and Eq. (8) hold if ξ = ξ . However, as we showin the SI, that the solution for ξ = ξ has the same form, andcan be obtained as the limit for ξ → ξ of Eq. (8). Again,the functional form for P ( v ) is the same in the 2HB and 1HBmodel, which are only differentiated by the dependence on F and [T] of the chemical rates. Analyses of experimental data.
We first analyzed the F =0 experimental data for Kin1 (12, 41) in order to obtainthe eight parameters at zero load by fitting Eq.(5) to therun length distribution, with the constraint that the averagevelocity, J + − J − = 132 . /s at [T] = 1mM (12), andthe ratio of forward over backward steps J + /J − = 221 at[T] = 10 µ M and [T] = 1mM (41). We also used the loaddependence of the average velocity at 1mM and 10 µ M ATPconcentration in Ref. 41 to obtain the parameters that dependon F and [T]. Following previous studies, we set F d = 3pN (15, 42) and | d + | + | d − | = 2 . k , K T , k +0 , and d + out of the eight Table 1. Extracted parameters for 2HB model. k . s − ) d + . nm ) k +0 . s − ) d − . nm ) k − . s − ) F d . pN ) γ . s − ) K T . µ M ) Table 2. Extracted parameters for 1HB model. k . s − ) d + . nm ) k +0 s − ) d − . nm ) k − . s − ) F d . pN ) γ . s − ) K T . µ M ) parameters in our model. The best fit parameters are listedin Table 1 and Table 2 for 2HB and 1HB model, respectively.It is worth pointing out that k +0 and k − for both the 1HBand 2HB models are fairly close to each other, and are inrough accord with our previous study that did not consider[T]-dependence (15). Similarly, the distances to the transitionstate when F = 0 ( d + and d − ) for both the schemes are notthat dissimilar (Table 1 and Table 2).In order to ascertain that our kinetic schemes for the 2HBand 1HB model provide a faithful description of the data ofMickolajczyk et al. (1) and Isojima et al. (2), we comparethe life-time of the 1HB [ τ HB = 1 / ( k + + k − + γ )] and 2HB( τ HB = 1 /k ) with the experimental measurements. As shownin Fig. 3 the agreement for both the scenarios is excellent,indicating that our theory captures the results of the exper-iments (1, 2) accurately. We hasten to emphasize that thedata from Mickolajczyk et al. and Isojima et al. were notused for fitting. The agreement is a genuine emergent featureof our kinetic model, which lends credence to the additionalpredictions made below. Velocity distribution is bimodal when F = 0 . We use the an-alytical solutions for P ( n ) [Eq. (5)] and P ( v ) [Eq. (8)] in orderto predict how the distributions of run length and velocitychange over a broad range of load and ATP concentrationsfor the two models (see Fig. 4). First, we note that the bi-modality of the velocity distribution, originally predicted byVu et al. (15), is evident at both high (1mM) and low (10 µ M)ATP concentrations. The peak at the negative v increases as F approaches F S . As the ATP concentration is lowered themotor slows down and the location of the peak of the velocitydistribution becomes closer to zero. Second, the P ( v )s at allvalues of F when [T] is 1mM are similar in the 1HB and 2HBscenarios (upper panel in Fig. 4), and hence cannot be used toeasily distinguish between them when the [T] is high. Althoughthe shape of P ( v ) does depend on the ATP waiting state atlow [T] (right panel in Fig. 4), which in principle amenable toexperimental test, the small qualitative difference may not besufficient to discriminate between the waiting states in prac-tice. To summarize, we showed that the bimodality of P ( v )is robust to changes in the concentration of ATP and modelused for the ATP waiting states. This provides experimentalflexibility in testing the predicted bimodality. Although theprediction of bimodal behavior as a function of [T] and F ismost interesting in its own right, it may be challenging to use | Takaki et al.
200 400 600 800 1000 1200 14000.00000.00050.00100.00150.00200.0025 v ( nm / s ) P ( v ) ( nm ) P ( L ) ( a )( b ) Fig. 2.
Simultaneous fits of P ( L ) ( L = sn with s = 8.2 nm) and P ( v ) at zeroload for Kin1 to the experimental data given in (12). Red circles are from experimentand the blue and green lines are results from our theory, for the 2HB and 1HBmodel, respectively. (a) Run length distribution. (b) Velocity distribution of Kin1. Thecomparison shows not only that the theory reproduces the measured data well butthe overlap of the blue and green lines shows that at zero load the differences in ATPwaiting states are not reflected in the distributions of the run length and velocity. P ( v ) as a probe to determine the nature of the ATP waitingstate in conventional kinesin. Randomness parameters are qualitatively different in the1HB and 2HB waiting states for ATP.
Fluctuation analysesin molecular motors are performed using the so-called chemi-cal and mechanical randomness parameters (13, 18, 43). Theformer describes the fluctuation of the enzymatic states of themotor, and is given by r C = ( h τ i − h τ i ) / h τ i . Here, τ is thedwell time of the motor at one site and the bracket denotes av-erage over an ensemble of motors. The mechanical randomnessparameter is given by, r M = lim t →∞ ( h n ( t ) i − h n ( t ) i ) / h n ( t ) i .It can be shown that r C = r M and bounded from 0 to 1 ifthere are no backward steps (44). However, it is possible that r M increases beyond 1 when load acts on the motor due to thepresence of backward steps. We found analytical expressionsfor r C and r M , which allowed us to compare the deviation ofthe two kinds of randomness parameter as the external loadincreases. We can recover r C from r M by using the relation r C = [(2 P + − r M − P + (1 − P + )] / (2 P + − , where P + is theprobability of forward stepping. We denote the chemical ran-domness parameter calculated from mechanical randomnessparameter given above as ¯ r C in order to differentiate it from r C ,which is not easy to measure experimentally (44).The relation-ship connecting r C and r M has been derived elsewhere (45, 46).In the SI, we provide an alternate method, which connectsbetween the chemical and mechanical randomness parameters.The chemical randomness parameter in our model is writtenas, r C = k + ( k + + k − + γ ) ( k + k + + k − + γ ) . [9] [ T ](µ M ) τ ( m s ) ⌧ HB (Theory) ⌧ HB (Theory) ⌧ HB (Experiment) ⌧ HB (Experiment) [ T ](µ M ) τ ( m s ) ( a ) ( b ) ⌧ HB (Theory) ⌧ HB (Theory) ⌧ HB (Experiment) ⌧ HB (Experiment) Fig. 3.
Mean dwell time for the 2HB state ( τ HB ) and 1HB state ( τ HB ) as afunction of ATP concentration at F = 0 . The upper panel is for the 2HB model[Fig.1(c)] and the lower panel is for the 1HB model [Fig.1(d)]. Red circles and bluesquares are taken from the experiment by Mickolajczyk et al. (1) (upper panel) andIsojima et al. (2) (lower panel). Lines are the theoretical predictions for the dwelltimes for 1HB state and 2HB state. Note that in (a) τ HB is [T] independent whereasin (b) τ HB does not depend on [T]. In order to calculate the moments needed to calculate r M , we first obtain the re-normalized probability distribution,¯ f ( n > , t ), for the position of the motor at time t on thetrack, ¯ f ( n, t ) = 1 C ∞ X l =0 γ √ π ( n + 2 l )! l ! e − ξ ξ t t n +2 l + k n +2 l +1 ( k + ) n + l ( k − ) l | ξ − ξ | n +2 l + I n +2 l + (cid:0) | ξ − ξ | t (cid:1) . [10] The normalization constant C , which accounts for the detach-ment of motors is obtained by summing over both positiveand negative values of n in the above equation (see SI fordetails). By computing the first and second moments of ¯ f for n at sufficiently long times, we can obtain an expressionfor the mechanical randomness parameter r M . Because r C in Eq.(9) depends on ATP, which occurs in different steps inthe 2HB and 1HB model [Fig. 1(c) and (d), respectively], thevariation of r C as a function of [T] could be used to assess thelikelihood of the two models.In Fig. 5 we plot the randomness parameters, r M , ¯ r C , and r C for the kinetic schemes in Fig. 1(c) and Fig. 1(d) as a func-tion of ATP concentration at different loads. The dependenceon ATP concentration of the mechanical randomness parame-ters for kinesin have been previously reported (13, 18, 47). Weplotted the randomness parameter obtained in the experimentby Visscher et al. (13) and Verbrugge et al. (47) in Fig. 5in order to assess if the theory captures the experimentalbehavior. It is clear that the theory and experiments agreeonly qualitatively with the trends being very similar. It isknown from even more complicated models that it is difficult Takaki et al.
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200 0 200 4000.00000.00050.00100.00150.00200.0025 v ( nm / s ) P ( v ) - - - - - -
100 0 100 200 3000.0000.0010.0020.0030.0040.0050.0060.007 v ( nm / s ) P ( v ) ( a )( b ) Fig. 4.
Velocity distributions for v = 0 predicted by our theory for different loadsand ATP concentrations. Lines are for the 2HB model [Fig.1(c)] and dashed lines arefor the 1HB model [Fig.1(d)]. Colors represent different load applied to kinesin. (a)Velocity distribution at 1mM ATP concentration. (b) Velocity distribution for 10 µ M ATPconcentration. to calculate with high accuracy the dependence of variousrandomness parameters on F and [T] (13, 48). Because therandomness parameter measures the inverse of the number ofrate-limiting states in the cycle, it is not unreasonable thatour model may overestimate the randomness parameter. Athigher forces our model for chemical randomness is in near-quantitative agreement, ( F = 6pN). In addition, we recoverthe trend observed in experiments ( F = 1pN). At intermediatevalues of the forces ( F = 4pN) the agreement is less accurate.Thus, we surmise that the agreement between theory and ex-periments is reasonable so that we can discuss the use of theseparameters in deciphering the ATP waiting state of kinesin.Remarkably, the dependence of the randomness parameterson [T] and F is dramatically different in the two models forthe ATP waiting states. In the 1HB model, the randomnessparameters decreases monotonically. In sharp contrast, if ATPbinds when both heads are engaged with the MT, we predicta non-monotonic function of [T] with a minimum occurringat near [T] ≈ µ M. This finding suggests an alternative,and perhaps a more straightforward way, of differentiatingbetween two types of waiting states for ATP. If the randomnessparameters ( r M and r C ) could be measured using the higherresolution single molecule experiments (2) as a function of [T]and F , then the timing of ATP binding to kinesin could beunambiguously determined.It is most interesting that at all values of F the model basedon the 2HB waiting state the randomness parameters has aclear minimum as the ATP concentration is changed whereas inthe 1HB model the decrease is monotonic and is almost flat as F increases. The difference can be appreciated by noting thatin the 2HB model the rate determining step for completing astep changes as [T] is increased from a low value. In particular,at low [T] the rate limiting step is the 2HB → → r C = x/k ) (1+ x/k ) where x = k + + k − + γ . Using the parameters in Table 1 we determinethat at low [T] with F = 0 ( xk (cid:29)
1) whereas at saturatingATP concentration xk ≈ . xk = 1.The values of [T] at which the randomness parameters are aminimum at different values of F may be estimated using thevalues in Table 1, which is roughly in accord with the resultsin Fig.5.In sharp contrast, in the 1HB model the 1HB → → F increases. This is because the 1HB → → → xk being less thanunity. As a consequence, the chemical randomness parameteris nearly monotonic and is close to unity at all values of F ,thus making r C almost independent of [T] (see Fig.5).It might be tempting to conclude based on the random-ness parameter at zero load reported in (47) [Fig. 5(a)] thatthere is a small dip around 100 µ M as predicted theoreticallyusing the 2HB model [Fig.1(c)]. Although not unambiguous,the randomness parameter with external loads measured byVisscher et al. (13) [Fig. 5(b)-(d)] apparently shows more orless a monotonic decease with increasing [T], which agreeswith the predictions of the 1HB model [Fig.1(d)]. We notethat the experiment at zero load [Fig. 5(a)] was conductedby using fluorescence microscopy and those at non-zero loadused optical trapping technique. Because of limited tempo-ral resolution in prior experiments, all the measurements ofrandomness parameter correspond to r M , the mechanical ran-domness parameter. With access to temporal resolution onthe order of tens of microseconds, it may be possible to di-rectly measure the chemical randomness parameter. For afuller understanding of mechano-chemistry of kinesin and inparticular how Kin1 waits for ATP, it is desirable to explorethe [T] and F dependence of chemical/mechanical parametersusing high resolution stepping trajectories. Discussion
We have introduced a simple model for stepping of conventionalkinesin on microtubule in order to propose single moleculeexperiments, which could be used to discriminate betweenthe waiting states for ATP binding to the leading head. Wederived analytical solutions for the run length and velocitydistributions and various randomness parameters as a functionof ATP-concentration and external resistive load. For both the1HB model and 2HB models P ( n ) is independent of [T], whichis in good agreement with experiments except at very low [T]concentrations, perhaps due to enhanced probability of spon-taneous detachment (13, 47). Therefore, although P ( n ) couldbe measured readily it cannot be easily used to distinguishbetween the two distinct waiting states. The distribution ofvelocity, which exhibits bimodal behavior at F = 0, is qual-itatively similar both at high and low ATP concentrations.The velocity distribution does differ quantitatively at low ATPconcentrations as F is varied [see Fig. 4(b)]. The most signifi- | Takaki et al. ant finding is that that the randomness parameters, whichcould be measured readily, shows qualitative differences as afunction of F and [T] between the 2HB and 1HB waiting statefor ATP. Predicted Bi-modality in the velocity distributionis independent of the ATP waiting states:
Since themean run length does not depend significantly on the ATPconcentration for Kin1 (13, 47) it follows that the mean posi-tion from which the motor detaches from the MT is roughly thesame irrespective of ATP concentrations. Thus, [T] would notaffect the spatial resolution needed to observe the predictedbimodality in the velocity distribution. However, since theaverage velocity of kinesin increases with [T], it would affectthe temporal resolution needed to validate the shape in P ( v ).We propose that it would be easier for experimentalists to ob-serve the theoretical prediction that P ( v ) is bimodal at lowerATP concentrations. This most interesting prediction, madea few years ago (15) without considering the [T]-dependencein contrast to this study, awaits experimental tests. Randomness parameters are dramatically differentbetween the two waiting states:
We predict that the [T]and F dependence of the randomness parameters, which isan estimate of the minimum number of rate limiting statesin kinesin, holds the key in assessing the relevance of the twowaiting states. Since the theory for both the 2HB and 1HBmodel consider only two states, the calculated randomnessparameters cannot be below 0.5. Therefore, it might be tempt-ing to conclude that our predictions may not be realizable inexperiments because it has been advocated that more than twostates might be needed to fit the experimental data (20, 21). However, we argue that the qualitative features of the [T]-dependence of the randomness parameter elucidated using ourtheory should be observable in experiments using the followingreasoning. Because kinesin has only one ATP-dependent rateper step and the rest of the rates do not depend on ATP,just as in our model, the change of randomness parameteras a function of [T] is only affected by the step that dependson ATP concentration. On the other hand, we compressedmany potentially relevant states into one internal state thatare unaffected by [T]. As a consequence, we expect that when[T] becomes large, our model might overestimate the values ofthe randomness parameters by a factor that is proportional tothe number of actual ATP-independent internal states. Indeed,if we shift our values for r in Fig.5(a) so that they match theexperimental values at high [T], we would attain an excellentagreement with the data. The presence of force might furthercomplicate the interplay between internal states. Nevertheless,the qualitative difference between the 1HB and 2HB modelshould be amenable to experimental verification. Therefore,we believe that accurate measurements of r M and r C usinghigh temporal resolution experiments will be most useful infilling a critical missing gap in the catalytic cycle of Kin1. Status of experiments and relation to theory:
Ran-domness parameters have been measured previously usingfluorescence microscopy (47) and optical trapping (13, 18).The experimental set up in (47) did not contain cargo whereasthe stepping trajectories in the optical trapping experimentswere measured by monitoring the time-dependent movementof a bead attached to the coiled-coil(13, 18). Both experi-ments from Hancock and coworkers (1) and Tomishige and - - [ T ](µ M ) r - - [ T ](µ M ) r - - [ T ](µ M ) r - - [ T ](µ M ) r F=0pN F=1pNF=4pN F=6pN ( a ) ( b )( c ) ( d ) Fig. 5.
Theoretical prediction of the ATP concentration dependence of the three randomness parameters, r M , r C , and ¯ r C at different external loads for the 2HB and 1HBmodel [Fig.1(c) and (d), respectively]. Filled squares, filled circles, and lines denote r M , ¯ r C , and r C , respectively. Red circles with error bar in (a) are the experimentallymeasured randomness parameter at F = 0 in (47). Red circles with error bar in (b)-(d) are the randomness parameters measured in (13); (b) for 1.05 pN, (c) for 3.59 pN, and(d) for 5.69 pN. As explained in the discussion section, the values of the randomness parameters in our schemes are always equal or grater than 0.5.Takaki et al. PNAS |
August 22, 2019 | vol. XXX | no. XX | oworkers (2), employ innovative experimental methods, whichare different from the techniques previously used to measurethe randomness parameters. These experiments also did nothave cargo but a large AuNP (with diameters between 20 to40 nm) was attached to different sites on one of the motorheads. The AuNP experiments should have sufficient tempo-ral and spatial resolution to extract both the mechanical andchemical randomness parameters as a function of ATP con-centration. The current iSCAT or experiments based on darkfield microscopy may not be able to measure the randomnessparameter as a function of F , which would require attachinga bead (cargo) that would not interfere with the dynamics ofAuNP. Nevertheless, measurements of randomness parametersusing the experimental constructs in (1, 2) as a function of[T] but with F = 0 can be made. Such studies are needed totest our predictions (Fig.5a), which would hopefully provideinsights into the ATP waiting state of kinesin. Mechano-chemistry of the backward step:
In ourmodel for the 1HB waiting state [Fig.1(d)], we assumed thatthe rate of the backward stepping depends on [T] in the samemanner as the rate for the forward step. It stands to rea-son that any step should consume ATP, and consequently k − should also depend on [T]. Indeed, it has been argued thatKin1 walks backwards by a hand-over-hand mechanism byhydrolyzing ATP in much the same as it does when moving for-ward (14, 41). The observation that the ratio of the probabilityof taking forward to backward steps as a function of F at two[T] concentrations (1 mM and 10 µ M) superimpose [see Fig.4bin (41)] lends support to the supposition that k − should alsodepend on [T]. Our 2HB and 1HB models [Fig.1(c) and (d),respectively], which consider ATP binding even for backwardsteps, leads to the prediction that both the run length and thefraction of forward step to backward step are independent of[T], as shown in the experiments (14, 41). In addition, severalof theoretical models have been proposed to rationalize the[T]-dependence of the backward step (14, 21, 23, 24, 49, 50).Therefore, our assumption that k − depends on [T] seemsjustifiable.However, the mechanism, especially in structural terms,of the backward step is not fully understood (14, 21, 49, 50).Therefore, it is important to entertain the possibility that k − has negligible dependence on [T]. Note that the magnitude of k − is non-negligible only in the presence of substantial load.At very low forces one could neglect the [T]-dependence of k − . Under these conditions the mechanisms for forward andbackward steps need not be the same.There are at least two possible pathways (see Fig.6) bywhich Kin1 could take backward steps. (I) Let us considerthat ATP binds to the LH in either 2HB state or 1HB stateand the TH detaches with bound ADP. In order for a backwardstep to occur, the TH has to release ADP and perform a "footstomp" (return to the starting position). Although to datethere is no evidence for either TH or LH foot stomping inKin1, they have been observed in Myosin V in the absenceof external load (51). The probability of foot stomping couldcertainly increases if F = 0, but is improbable in the absenceof load. If stomping were to occur, then both the heads wouldbe bound to the MT with the LH containing ATP (third stepin pathway I in Fig.6). After TH stomping, ATP should behydrolyzed and the inorganic phosphate released from theLH, which would lead to backward stepping. This pathway results in identical [T] dependence for forward and backwardsteps. Consequently, the [T] independent characteristics ofKin1, such as P ( n ), can be explained by this scenario. (II)Let us consider another possibility for backward steps. BeforeATP binds to the LH in either 1HB state or 2HB state, ADPis released from TH, leading to 2HB state with both the headsbeing nucleotide free pathway II in Fig.6). For backwardstate to occur from this state, the LH should detach from the2HB state either spontaneously or by binding ATP. The latterevent, which would induce neck linker docking, and hencepropel the TH forward would tend to suppress the probabilityof backward steps. If the former were to occur then it mightbe possible, especially if F = 0, that k − might not depend on[T].The theory developed based on the scheme in Fig.1(d) doesnot account for the possibility that backward step rate maynot depend on [T]. For completeness, we created in the SI avariant of the 1HB model, corresponding to scenario (II), bysetting k − in Fig.1(d) to be independent of [T]. The results inthe SI show that regardless of the dependence or independenceof k − on [T] the qualitative differences in the randomness pa-rameters as a function of F and [T] between the 1HB and 2HBmodel remain. Thus, the theoretical predictions are robust,suggesting that high temporal resolution experiments thatmeasure randomness could be used to discriminate betweenthe two waiting states for ATP. Conclusion
It has been challenging to decipher how exactly kinesin waitsfor ATP to bind to the leading head. Recent experiments havearrived at contradictory conclusions using similar experimentaltechniques. Although one cannot rule out the possibility thatdifferent kinesin constructs and the location of attachment ofthe gold nanoparticle used in these experiments might lead todifferent stepping trajectories, it is important to consider thetheoretical consequences of the two plausible waiting states ofkinesin. To discriminate between the 1HB and 2HB waitingstates, we developed simple models, allowing us to calculateanalytically and fairly accurately a number of measurablequantities. The theory predicts that there ought to be qualita-tive differences in the randomness parameters as a function ofload and ATP concentration. Although the force dependenceof the randomness parameters have been previously measuredusing optical trap techniques, it would be most interesting torepeat these measurements using the constructs used in themost recent experiments (1, 2). In addition, measurements ofthe load dependence of the randomness parameters using acombination of dark field microscopy methods in combinationwith optical traps would be most illuminating to verify manyof the predictions outlined here.
Materials and Methods.
We created two stochastic kinetic mod-els in order to calculate a number of quantities associated withthe stepping kinetics of conventional kinesin. The sketch ofthe 1HB and 2HB models and the pathways leading fromthe resting state to the target binding states along with therates and [T]-dependence are given in Fig. 1. The model, ageneralization of the one introduced previously (15) in orderto include the important aspect of [T]-dependence, can besolved exactly, thus allowing us to calculate P ( v ) and the dif-ferent randomness parameters for the two different scenarios | Takaki et al. T T φ T φ T φ D φ D φ T φ T φφ DP D φ T φ T φφ DP T φ T φφ φ D φ D φφφ D φφ T (I)(II) [T][T] [T] Stomp S t o m p Fig. 6.
Plausible backward step mechanisms for kinesin. Upper panel corresponds to pathway (I) explained in the discussion section. In this case the [T] dependence isidentical to forward stepping. Lower panel is pathway (II) in which [T] dependence could be different from the forward stepping. T, D, DP, and φ stands ATP, ADP, (ADP +phosphate), and no nucleotide state, respectively. The two pathways partially adapted from Ref.(49). In the upper panel, after ATP binding to the leading head (shown in red)the the neck linker docks. Consequently, the backward step along this pathway would only be possible at higher loads. for the waiting states for ATP binding (see SI appendix fordetails). Despite the simplicity, we show in the SI that themodel does quantitatively reproduce the experimentally mea-sured [T]-dependent force-velocity relation using physicallyreasonable parameters for the rates describing the two schemes[Fig. 1(c) and (d)]. ACKNOWLEDGMENTS.
We are grateful to Ahmet Yildiz andWilliam Hancock for their interest and useful comments. This workwas supported by National Science Foundation Grant CHE-1900093.Additional support was provided by Collie-Welch Reagents ChairF-0019.
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Nature | Takaki et al. upplementary InformationHow kinesin waits for ATP affects the nucleotide and loaddependence of the stepping kinetics R. Takaki
Department of Physics, The University of Texas at Austin, Austin, TX 78712
M. L. Mugnai, Y. Goldtzvik, and D. Thirumalai
Department of Chemistry, The University of Texas at Austin, Austin, TX 78712 (Dated: August 22, 2019) a r X i v : . [ q - b i o . S C ] A ug ONTENTS
I. Derivation of the run length distribution 3II. Derivation of the velocity distribution 3III. Randomness parameter 8A. Chemical randomness parameter, r C r M : k − is independent of [T] 20VI. Fitting theory to experimental data 21References 232 . DERIVATION OF THE RUN LENGTH DISTRIBUTION The summation in Eq.(4) in the main text, P ( n ) = ∞ X m,l =0 ( m + l )! m ! l ! (cid:16) J + J T (cid:17) m (cid:16) J − J T (cid:17) l (cid:16) J γ J T (cid:17) δ m − l,n , (S1)can be carried out for n >
0, leading to, P ( n >
0) = (cid:16) J + J T (cid:17) n (cid:16) J γ J T (cid:17) ∞ X l =0 (cid:16) J + J − J T (cid:17) l (2 l + n )( n + l )! l != (cid:16) J + J T (cid:17) n (cid:16) J γ J T (cid:17) F (cid:16) n , n n ; 4 J + J − J T (cid:17) , (S2)where F is Gaussian hypergeometric function. By using the special case of F (page 556of [1]), we obtain the following expression for the run length distribution, P ( n ≷
0) = (cid:16) J ± J T + p J T − J + J − (cid:17) | n | J γ p J T − J + J − . (S3)In the above equation n is non-dimensional quantity which denotes the position of the motoron the track. Notice that Eq.(S3) is independent of ATP concentration, which is in accordwith experiments [2, 3] as long as ATP concentration exceeds ≈ µ M. At lower ATPconcentration, the mean run length does moderately depend on [T]. This suggests that atvery low [T] the motor head could enter the vulnerable 1HB state spontaneously. In such astate the LH could detach before ATP binds to it. This would occur if the binding time forATP (inverse of a pseudo first order constant) is greater than the detachment time.
II. DERIVATION OF THE VELOCITY DISTRIBUTION
In our model (Fig.1(c) and (d) in the main text), the probability distribution of forwardstep, backward step, and detachment at time t , and the corresponding Laplace transform3 L ( f ) = R ∞ e − st f ( t ) dt ) are given by the following expressions, f + ( t ) = Z t dt k e − kt k + e − ( k + + k − + γ )( t − t ) = kk + k + + k − + γ − k (e − kt − e − ( k + + k − + γ ) t )˜ f + ( s ) = kk + ( s + k )( s + k + + k − + γ ) ; (S4)and f − ( t ) = Z t dt k e − kt k − e − ( k + + k − + γ )( t − t ) = kk − k + + k − + γ − k (e − kt − e − ( k + + k − + γ ) t )˜ f − ( s ) = kk − ( s + k )( s + k + + k − + γ ) ; (S5) f γ ( t ) = Z t dt k e − kt γ e − ( k + + k − + γ )( t − t ) = kγk + + k − + γ − k (e − kt − e − ( k + + k − + γ ) t )˜ f γ ( s ) = kγ ( s + k )( s + k + + k − + γ ) . (S6)The probability that the motor takes m forward steps and l backward steps before detachingat time t can be written as, f ( m, l, t ) = ( m + l )! m ! l ! Z t dt m + l Z t m + l dt m + l − · · · Z t dt Z t dt m Y i =1 f + ( t i − t i − ) m + l Y i = m +1 f − ( t i − t i − ) f γ ( t − t m + l ) . (S7)The factorial term in Eq.(S7) accounts for the number of ways in which the motor can take m forward steps and l backward steps for a net displacement of m − l . In Laplace space,4q.(S7) is a multiplication of all the dwell time distributions up to the time of detachment,˜ f ( m, l, s ) = ( m + l )! m ! l ! ( ˜ f + ( s )) m ( ˜ f − ( s )) l ˜ f γ ( s )= ( m + l )! m ! l ! k m + l +1 ( k + ) m ( k − ) l γ s + k ) m + l +1 ( s + k + + k − + γ ) m + l +1 . (S8)In order to obtain f ( m, l, t ) by inverse Laplace transform, we need to evaluate the followingintegral, 12 πi Z c + i ∞ c − i ∞ s + ξ ) n +1 ( s + ξ ) n +1 e st ds, (S9)where we set m + l = n , ξ = k , and ξ = k + + k − + γ . This can be done using the standardresidue theorem. For the generic case of ξ = ξ , there are two distinct poles of order n + 1.Res h s + ξ ) n +1 ( s + ξ ) n +1 e st i(cid:12)(cid:12)(cid:12) s = − ξ = 1 n ! d n ds n h s + ξ ) n +1 e st i(cid:12)(cid:12)(cid:12) s = − ξ = 1 n ! n X p =0 ( − p (cid:18) np (cid:19) t n − p e − ξ t ( − ξ + ξ ) n + p +1 ( n + p )! n ! , (S10)Res h s + ξ ) n +1 ( s + ξ ) n +1 e st i(cid:12)(cid:12)(cid:12) s = − ξ = 1 n ! d n ds n h s + ξ ) n +1 e st i(cid:12)(cid:12)(cid:12) s = − ξ = 1 n ! n X p =0 ( − p (cid:18) np (cid:19) t n − p e − ξ t ( − ξ + ξ ) n + p +1 ( n + p )! n ! . (S11)Thus, by combining Eqs.(10.47.9), (10.49.1), and (10.49.12) of the NIST handbook of Math-ematics [4], the inverse Laplace transform in Eq.(S9) becomes,( S
9) = 1 n ! n X p =0 ( − p (cid:18) np (cid:19) t n − p e − ξ t ( − ξ + ξ ) n + p +1 ( n + p )! n ! + 1 n ! n X p =0 ( − p (cid:18) np (cid:19) t n − p e − ξ t ( − ξ + ξ ) n + p +1 ( n + p )! n != t n √ πn ! e − ξ ξ t p ( ξ − ξ ) t ( ξ − ξ ) n +1 K n + ( ξ − ξ t ) + t n √ πn ! e − ξ ξ t p ( ξ − ξ ) t ( ξ − ξ ) n +1 K n + ( ξ − ξ t ) , (S12)5here K is the modified Bessel function of the second kind.Hence, Eq.(S7) becomes, f ( m, l, t ) = γm ! l ! 1 √ π e − ξ ξ t t m + l k m + l +1 ( k + ) m ( k − ) l h p ( ξ − ξ ) t ( ξ − ξ ) m + l +1 K m + l + (cid:0) ξ − ξ t (cid:1) + p ( ξ − ξ ) t ( ξ − ξ ) m + l +1 K m + l + (cid:0) ξ − ξ t (cid:1)i . (S13)We can simplify the above expression by employing the following identity for modified Besselfunction given in Eq.(10.34.2) in [4], and rewrite it as, K n + ( − x ) = − i (cid:0) πI n + ( x ) + ( − n K n + ( x ) (cid:1) ( x > , (S14)where I is the modified Bessel function of the first kind. Therefore, we further simplifyEq.(S13) as follows.For ξ − ξ > f ( m, l, t ) = γm ! l ! √ π e − ξ ξ t t m + l k m + l +1 ( k + ) m ( k − ) l ( ξ − ξ ) m + l +1 p ( ξ − ξ ) t I m + l + (cid:0) ξ − ξ t (cid:1) . (S15)If ξ − ξ > f ( m, l, t ) = γm ! l ! √ π e − ξ ξ t t m + l k m + l +1 ( k + ) m ( k − ) l ( ξ − ξ ) m + l +1 p ( ξ − ξ ) t I m + l + (cid:0) ξ − ξ t (cid:1) . (S16)Both cases are written as, f ( m, l, t ) = γ √ πm ! l ! e − ξ ξ t t m + l k m + l +1 ( k + ) m ( k − ) l | ξ − ξ | m + l +1 p | ξ − ξ | t I m + l + (cid:0) | ξ − ξ | t (cid:1) . (S17)6e finally obtain the expression for the velocity distribution.For m − l > P ( v >
0) = ∞ X m,lm>l Z ∞ f ( m, l, t ) δ ( v − m − lt ) dt = ∞ X m,lm>l m − lv γ √ πm ! l ! e − ξ ξ m − lv (cid:0) m − lv (cid:1) m + l + k m + l +1 ( k + ) m ( k − ) l | ξ − ξ | m + l + I m + l + (cid:0) | ξ − ξ | m − lv (cid:1) . (S18)For m − l < P ( v <
0) = ∞ X m,ll>m Z ∞ f ( m, l, t ) δ ( v − m − lt ) dt = ∞ X m,ll>m l − mv γ √ πm ! l ! e − ξ ξ m − lv (cid:0) m − lv (cid:1) m + l + k m + l +1 ( k + ) m ( k − ) l | ξ − ξ | m + l + I m + l + (cid:0) | ξ − ξ | m − lv (cid:1) . (S19)Where ξ = k and ξ = k + + k − + γ .Let us now consider the case ξ = ξ . Let ξ = ξ ≡ ξ , then we need to evaluate thefollowing integral, 12 πi Z c + i ∞ c − i ∞ s + ξ ) n +1) e st ds. (S20)In this case, we have a single pole of order 2( n + 1). The result for f ( m, l, t ) becomes, f ( m, l, t ) = ( m + l )! m ! l ! k m + l +1 ( k + ) m ( k − ) l γ t m + l )+1 e − ξt (2( m + l ) + 1)! , (S21)leading to P ( v >
0) = ∞ X m,lm>l m − lv ( m + l )! m ! l ! k m + l +1 ( k + ) m ( k − ) l γ (2( m + l ) + 1)! (cid:16) m − lv (cid:17) m + l )+1 e − ξ m − lv , (S22)7 ( v <
0) = ∞ X m,ll>m l − mv ( m + l )! m ! l ! k m + l +1 ( k + ) m ( k − ) l γ (2( m + l ) + 1)! (cid:16) m − lv (cid:17) m + l )+1 e − ξ m − lv . (S23)Note that Eqs.(S21)-(S23) could also be obtained by taking the limit for ξ → ξ in Eqs.(S17)-(S19), knowing that the limit for small argument of the modified Bessel function of the firstkind is given in Eq.(10.30.1) of [4].For the kinetic scheme in Fig.1(c) and (d) in the main text, the expressions for thevelocity distributions are identical. However, the rate k , k + , k − , and γ depend on ATPconcentration in a different manner for the two models, which describe two distinct waitingstates of kinesin for ATP. III. RANDOMNESS PARAMETER
Randomness parameter, which in some sense is easier to measure in experiments, is auseful way to estimate the number of rate limiting steps of molecular motors [5, 6]. Wehere discuss two types of randomness parameters for our models in the main text, chemicalrandomness r C and mechanical randomness r M , which are connected by a relation denotedby ¯ r C as shown below. A. Chemical randomness parameter, r C Chemical randomness is given by the following expression, r C = h τ i − h τ i h τ i , (S24)where τ is the dwell time of the motor at a given site and the bracket denotes an averageover an ensemble of motors. Dwell time distributions for stepping forward ( + ) or backward( − ) at a site at time t in our models [Fig.1(c) and (d) in the main text] are given by, f ± ( t ) = Z t dt k e − kt k ± e − ( k + + k − + γ )( t − t ) = kk ± k + + k − + γ − k (e − kt − e − ( k + + k − + γ ) t ) . (S25) h τ ± i = R ∞ τ ± f ± ( τ ± ) dτ ± R ∞ f ± ( τ ± ) dτ ± = k + k + + k − + γk ( k + + k − + γ ) , (S26) h τ ± i = R ∞ τ ± f ± ( τ ± ) dτ + R ∞ f ± ( τ ± ) dτ ± =2 k + k ( k + + k − + γ ) + ( k + + k − + γ ) k ( k + + k − + γ ) . (S27) Thus, ( h τ i − h τ + i ) / h τ + i = ( h τ − i − h τ − i ) / h τ − i , which allows us to express r C as r C = k + ( k + + k − + γ ) ( k + k + + k − + γ ) . (S28) B. Mechanical randomness parameter, r M We define r M as, r M = lim t →∞ h n ( t ) i − h n ( t ) i h n ( t ) i , (S29)where n ( t ) is the position of the motor at time t . In our model, we can obtain the expressionfor the probability distribution that the motor is at site n at time t , which is needed tocalculate the moments in (S29).Using (S17) we obtain the probability that the motor takes n = ( m − l ) , f ( n, t ) = X m,l δ m,n + l f ( m, l, t )= ∞ X l =0 γ √ π ( n + 2 l )! l ! e − ξ ξ t t n +2 l + k n +2 l +1 ( k + ) n + l ( k − ) l | ξ − ξ | n +2 l + I n +2 l + (cid:0) | ξ − ξ | t (cid:1) . (S30)However, as written the sum (S30) accounts for contributions from motors that have de-tached from the track before sufficiently long time t has elapsed. The appropriate probabilitydistribution to get the moments in Eq.(S29) is the re-normalized probability distribution ateach time t , which accounts only for the motors which stay on the track for a long time t .We denote this probability distribution as ¯ f , which is defined as,9 IG. S1. Relaxation of mechanical randomness parameter at F = 0pN (blue), F = 4pN (green),and F = 6pN (red). [T]=1mM in all cases. (a) 2HB model. (b) 1HB model. In all cases t = 0 . r M to have a plateau value. ¯ f ( n, t ) = 1 C ∞ X l =0 γ √ π ( n + 2 l )! l ! e − ξ ξ t t n +2 l + k n +2 l +1 ( k + ) n + l ( k − ) l | ξ − ξ | n +2 l + I n +2 l + (cid:0) | ξ − ξ | t (cid:1) , (S31)where C = ∞ X l =0 ∞ X n = −∞ γ √ π ( n + 2 l )! l ! e − ξ ξ t t n +2 l + k n +2 l +1 ( k + ) n + l ( k − ) l | ξ − ξ | n +2 l + I n +2 l + (cid:0) | ξ − ξ | t (cid:1) . (S32)We used the distribution Eq.(S31) to obtain the first and second moment of n for thecalculation of mechanical randomness parameter. In practice, we calculated the first andsecond moments h n ( t ) i and h n ( t ) i at t = 0 . h n ( t ) i to decaysignificantly. The double summation in Eq.(S32) should include enough terms to ensureconvergence of h n ( t ) i and h n ( t ) i . We truncated the summation at 30 and 130 for l and n ,respectively. Needless to say that if t is extended beyond 0 .
5s then a larger number of termswill have to be calculated to obtain converged results.If we denote ¯ r C as the chemical randomness parameter, which takes backward steps intoaccount, we found that, 10 r C = (2 P + − r M − P + (1 − P + )(2 P + − . (S33)Thus, ¯ r C can be calculated from mechanical randomness parameter by taking into accountbackward steps. The derivation of this equation can be found in the literature in a differentcontext [7, 8]. We show that in the next section this relation can be derived by includingbackward steps in the work by Schnitzer and Block [6]. In order to calculate ¯ r C we need tocompute the probability of forward step, P + . We assume P + + P − = 1 for all times untilthe motor detaches.Thus, in our model, P + = R ∞ f + ( τ + ) dτ + R ∞ f + ( τ + ) dτ + + R ∞ f − ( τ − ) dτ − = k + k + + k − . (S34)Using r M and P + , we are able to calculate ¯ r C . C. Derivation of mechanical randomness parameter with backward steps
We derive Eq.(S33) by incorporating backward steps into the work by Schnitzer andBlock [6]. We denote the dwell time distribution corresponding to forward and backwardsteps as f + ( t ) and f − ( t ), respectively. Let g ( m, l, t ) be the probability distribution of taking m forward steps and l backward steps before time t . The Laplace transform of g ( m, l, t ), iswritten as follows,˜ g ( m, l, s ) = ( m + l )! m ! l ! ( ˜ f + ( s )) m ( ˜ f − ( s )) l − ˜ f + ( s ) − ˜ f − ( s ) s . (S35)The last term in (S35) is the Laplace transform of 1 − R t ( f + ( t ) + f − ( t )) dt , means neitherforward nor backward step occurs in the last time step. For n ≡ m − l ≥ g + ( n, s ) = X m,l δ n,m − l ˜ g ( m, l, s )=[ ˜ f + ( s )] n F ( 1 + n , n , n ; 4 ˜ f + ( s ) ˜ f − ( s )) 1 − ˜ f + ( s ) − ˜ f − ( s ) s . (S36)11sing the special case of Hypergeometric function (page 556 of [1]) we obtain,˜ g + ( n, s ) = " f + ( s )1 + q − f + ( s ) ˜ f − ( s ) n q − f + ( s ) ˜ f − ( s ) 1 − ˜ f + ( s ) − ˜ f − ( s ) s . (S37)For n ≤
0, a similar procedure leads to,˜ g − ( n, s ) = " f − ( s )1 + q − f + ( s ) ˜ f − ( s ) − n q − f + ( s ) ˜ f − ( s ) 1 − ˜ f + ( s ) − ˜ f − ( s ) s . (S38)Thus,˜ g ± ( n, s ) = (cid:18) f ± ( s )1 + q − f + ( s ) ˜ f − ( s ) (cid:19) | n | q − f + ( s ) ˜ f − ( s ) 1 − ˜ f + ( s ) − ˜ f − ( s ) s . (S39)The first and second moment of n are defined as, h n ( s ) i = ∞ X n =0 n ˜ g + ( n, s ) + X n = −∞ n ˜ g − ( n, s ) ≡h n ( s ) i + + h n ( s ) i − h n ( s ) i = ∞ X n =0 n ˜ g + ( n, s ) + X n = −∞ n ˜ g − ( n, s ) ≡h n ( s ) i + + h n ( s ) i − . (S40)It is convenient to use the generating function of ˜ g ± ( n, s ) to calculate the moments, and itis a simple geometric sum given by, Z + ( x, s ) = ∞ X n =0 ˜ g + ( n, s ) x n = 11 − f + ( s ) x √ − f + ( s ) ˜ f − ( s ) q − f + ( s ) ˜ f − ( s ) 1 − ˜ f + ( s ) − ˜ f − ( s ) s . (S41)12he average number of forward step is calculated using the generating function as, h n ( s ) i + = ∂Z + ∂x (cid:12)(cid:12)(cid:12) x =1 = 2 ˜ f + ( s ) h q − f + ( s ) ˜ f − ( s ) iq − f + ( s ) ˜ f − ( s ) (cid:20) − f + ( s ) + q − f + ( s ) ˜ f − ( s ) (cid:21) − ˜ f + ( s ) − ˜ f − ( s ) s . (S42)The second derivative of the generating function leads to, h n ( s ) i + − h n ( s ) i + = ∂ Z + ∂x (cid:12)(cid:12)(cid:12) x =1 = 8[ ˜ f + ( s )] h q − f + ( s ) ˜ f − ( s ) iq − f + ( s ) ˜ f − ( s ) (cid:20) − f + ( s ) + q − f + ( s ) ˜ f − ( s ) (cid:21) − ˜ f + ( s ) − ˜ f − ( s ) s . (S43)Thus, h n ( s ) i + = 2 ˜ f + h q − f + ( s ) ˜ f − ( s ) ih f + ( s ) + q − f + ( s ) ˜ f − ( s ) iq − f + ( s ) ˜ f − ( s ) (cid:20) − f + ( s ) + q − f + ( s ) ˜ f − ( s ) (cid:21) − ˜ f + ( s ) − ˜ f − ( s ) s . (S44)In a similar manner, h n ( s ) i − = − f − ( s ) h q − f + ( s ) ˜ f − ( s ) iq − f + ( s ) ˜ f − ( s ) (cid:20) − f − ( s ) + q − f + ( s ) ˜ f − ( s ) (cid:21) − ˜ f + ( s ) − ˜ f − ( s ) s , (S45) h n ( s ) i − = 2 ˜ f − h q − f + ( s ) ˜ f − ( s ) ih f + ( s ) + q − f + ( s ) ˜ f − ( s ) iq − f + ( s ) ˜ f − ( s ) (cid:20) − f − ( s ) + q − f + ( s ) ˜ f − ( s ) (cid:21) − ˜ f + ( s ) − ˜ f − ( s ) s . (S46)13y substituting the above expressions in Eq.(S40) we obtain, h n ( s ) i = 2 h q − f + ( s ) ˜ f − ( s ) iq − f + ( s ) ˜ f − ( s ) 1 − ˜ f + ( s ) − ˜ f − ( s ) s " ˜ f + ( s ) h − f + ( s ) + q − f + ( s ) ˜ f − ( s ) i − ˜ f − ( s ) h − f − ( s ) + q − f + ( s ) ˜ f − ( s ) i , (S47) h n ( s ) i = 2 h q − f + ( s ) ˜ f − ( s ) iq − f + ( s ) ˜ f − ( s ) 1 − ˜ f + ( s ) − ˜ f − ( s ) s " ˜ f + ( s ) h f + ( s ) + q − f + ( s ) ˜ f − ( s ) ih − f + ( s ) + q − f + ( s ) ˜ f − ( s ) i + ˜ f − ( s ) h f − ( s ) + q − f + ( s ) ˜ f − ( s ) ih − f − ( s ) + q − f + ( s ) ˜ f − ( s ) i . (S48) We express ˜ f ± ( s ) using Taylor expansion,˜ f + ( s ) = P + ∞ X k =0 h τ k + i ( − s ) k k !˜ f − ( s ) = P − ∞ X k =0 h τ k − i ( − s ) k k ! . (S49)The moments of ˜ f ± ( t ) are given by, h τ k + i = R ∞ τ k + f + ( τ + ) dτ + R ∞ f + ( τ + ) dτ + = R ∞ τ k + f + ( τ + ) dτ + P + = ( − k d k ˜ f + ( s ) ds k P + , h τ k − i = R ∞ τ k − f − ( τ − ) dτ − R ∞ f − ( τ − ) dτ − = R ∞ τ k − f − ( τ − ) dτ − P − = ( − k d k ˜ f − ( s ) ds k P − . (S50)14rom here we assume the relation P + + P − = 1. Substituting (S49) into (S47) and (S48)with the condition either < P + ≤ ≤ P + < both yield, h n ( s ) i = 2 P + − (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) s + − P h τ + i + 2(1 − P + ) h τ − i + P + (2 P + − h τ i + (2 P + − − P + ) h τ − i (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) s + O (1) , (S51)and h n ( s ) i = 2(2 P + − (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) s + − P (8 P + − h τ + i + (1 − P + ) (8 P + − h τ − i + 2(2 P + − (cid:0) (1 − P + ) h τ − i + P + h τ i (cid:1) + 2(1 − P + ) P + h τ + ih τ − i (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) s + O ( s − ) . (S52) After inverse Laplace transform, h n ( t ) i = 2 P + − (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) t + − P h τ + i + 2(1 − P + ) h τ − i + P + (2 P + − h τ i + (2 P + − − P + ) h τ − i (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) + O ( t − ) , (S53)and h n ( t ) i − h n ( t ) i = P + (2 P + − h τ i − P (4 P + − h τ + i (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) t + (1 − P + )(2 P + − h τ − i − (1 − P + ) (1 − P + ) h τ − i (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) t + 2(1 − P + ) P + h τ + ih τ − i (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) t + O (1) . (S54)15ow we compute the randomness parameter as r M = lim t →∞ h n ( t ) i − h n ( t ) i h n ( t ) i = P + (2 P + − h τ i − P (4 P + − h τ + i (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) (2 P + − − P + )(2 P + − h τ − i − (1 − P + ) (1 − P + ) h τ − i (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) (2 P + − − P + ) P + h τ + ih τ − i (cid:0) P + h τ + i + (1 − P + ) h τ − i (cid:1) (2 P + − . (S55)For the special case, namely h τ + i = h τ − i and h τ i = h τ − i , r M in the above equation reducesto the following expression, r M = (2 P + − h τ i − (8 P − P + + 1) h τ i ( − P + ) h τ i . (S56)Manipulation of Eq.( S
56) leads to, r C = (2 P + − r M − P + (1 − P + )(2 P + − ≡ ¯ r C , (S57)where r C = h τ i−h τ i h τ i . Note that r C = r M if P + is unity, which holds in the absence ofbackward steps. 16 V. TWO MODELS FOR HOW KINESIN WAITS FOR ATP DP D DP φ D T D
DP DP k k k k k k k D OUT DP D DP T DP φ D D k k k k k k k OUT T φ φ T T φ FIG. S2. A sketch showing all the relevant rates involved in the stepping of kinesin. The upper(lower) panel shows the 2HB (1HB) waiting state. In the 2HB model , ATP binds to the LH whenboth heads are bound to the microtuble (MT). In the 1HB model ATP binding occurs only afterthe trailing head detaches from the MT. At extremely low ATP concentration the 1HB model ismore likely.
As explained in the main text, the two models (Fig.S3) have been proposed for the waitingstates of kinesin for ATP binding. In the 2HB model, ATP binds to the leading head whenboth the heads are bound to the MT (upper panel in Fig.S2). In contrast, Isojima et al. have suggested, based on dark field microscopy, that ATP binds only after the trailing headdetaches leading to the so-called vulnerable state (Fig.S3 bottom panel). In order to simplifythe kinetic scheme, we merge the 4 states in Fig.S2 into two states shown in Fig.S3 in order17 J HB ! HB J HB ! HB i i+1 i-1 OUT
FIG. S3. Simplified two states model. to calculate the net flux of transition from 2HB state to 1HB state ( J HB → HB ), forwardstep ( J HB → HB ), and detachment ( J γ ). The simplification allows us to obtain closed formexpressions for J HB → HB , J HB → HB , and J γ .In the following calculations, we employ method pioneered by Hill [9, 10]. We may designstate 3 to be the absorbing state for the transition from 2HB to 1HB (state 2 to state 3),state 1 to be the absorbing state for the transition from 1HB to 2HB (state 4 to state1). In addition, detachment is also an absorbing state when kinesin is in the 1HB state.By obtaining the stationary solution of the following sets of master equations, with theconditions P + P = 1 and P + P = 1, dP dt = − k P + ( k + k ) P ,dP dt = k P − ( k + k ) P , (S58) dP dt = − k P + ( k + k + γ ) P ,dP dt = k P − ( k + k + γ ) P , (S59)18e obtain, J HB → HB = k k k + k + k ,J HB → HB = k k γ + k + k + k ,J γ = γk γ + k + k + k , (S60)ATP binds with the rate k in the 2HB waiting state (upper panel in Fig.S2), which weparametrize as k = [T] k , where [T] is the ATP concentration. On the other hand, ATPbinding occurs with rate k in the 1HB model with detached TH. Thus, we include ATPdependence in k = [T] k . With these assumptions, the expressions for the fluxes are givenby, J HB → HB = k [T] k k + k + [T] k ,J HB → HB = k k γ + k + k + k ,J γ = γk γ + k + k + k , (S61)in the 2HB mdoel. The analogous expressions in the 1HB model are, J HB → HB = k k k + k + k ,J HB → HB = [T] k k γ + [T] k + k + k ,J γ = γ [T] k γ + [T] k + k + k , (S62)Thus, the Michaelis-menten (MM) kinetics naturally arises from this coarse grained proce-dure in J HB → HB (Eq.S61) and J HB → HB (Eq.S62). Since the MM constant in J HB → HB and J γ for the 1HB waiting model are identical, we used the functional form k + = k +0 [T] K T +[T] e − βF d + , k − = k − [T] K T +[T] e βF d − , and γ = γ [T] K T +[T] e F/F d . Using identical MM constantin k + , k − , and γ in 1HB waiting model leads to ATP independent ratio of probability offorward step, backward step, and detachment. This is realized in 2the HB waiting model aswell, and is in accord with experimental observation [11, 12].19 . s − ) d + . k +0 . s − ) d − . k − . s − ) F d . γ . s − ) K T . µ M) TABLE S1. Extracted parameters for the variant of 1HB model
V. VARIANT OF 1HB MODEL : k − IS INDEPENDENT OF [T]
It appears logical that the backward step should be the reverse of the forward step,which implies that it too should occur by a hand-over-hand mechanism with the rate beingdependent on ATP. For reasons discussed in the main text, depending on the pathway thatkinesin takes to go take a backward step, it is possible that rate of backward step k − doesnot depend on ATP binding. Therefore, we created a variant of the 1HB model [Fig.1(c)in the main text] in which k − is independent of [T]. The load dependence is identical tothe original 1HB model, namely k − = k − e βF − d . We followed the same procedure describedin the main text to obtain the parameters for the variant of 1HB model, which are listedin Table.S1. Interestingly, the values of the many parameters extracted using the 1HB andvariant 1HB models are not that dissimilar (compare Table.S1 and Table 2 in the main text).We show in Fig.S4 that our main prediction about the qualitative behavior of randomnessparameters are robust: the randomness parameters for the variant of 1HB waiting modelalso show monotonic decrease as [T] is increased. Thus, regardless of the mechanism of thebackward step we conclude that measurements of the randomness parameter as a function ofload and ATP concentration using currently available high temporal resolution experimentsshould resolve the nature of waiting states for ATP.20 IG. S4. Theoretical predictions for the ATP concentration dependence of the three randomnessparameters, r M , r C , and ¯ r C at different external loads for the 2HB model (blue) and the variantof 1HB model (green). Filled circles, filled squares and lines denote r M , ¯ r C , and r C , respectively.Red circles with error bar in (a) are the experimentally measured randomness parameter at F = 0in [2]. Red circles with error bar in (b)-(d) are the randomness parameters measured in [3]; (b)for 1.05 pN, (c) for 3.59 pN, and (d) for 5.69 pN. As explained in the discussion section in themain text, randomness parameters in our schemes are always equal or grater than 0.5. The resultsfor the 1HB model are obtained by assuming that the rate for the backward step is a constantindependent of the ATP concentration. VI. FITTING THEORY TO EXPERIMENTAL DATA
In order to obtain the parameters for the model, we analyzed the distribution of runlength and velocity at F = 0 using the data reported in Ref.[13]. For Kin1, the measuredmean velocity at 0 load is 1089 nm/s, which implies J + − J − = 132 . J + /J − is not given in Ref.[13], we used the value of J + /J − obtained in Ref.[11], J + /J − = 221. Weset F d = 3pN [14, 15] and used the constraint | d + | + | d − | = 2 . IG. S5. Fits of the of average velocity as a function of load to the experiment [11]. (a) 2HBwaiting model [Fig.1(c) in the main text]. (b) 1HB waiting model [Fig.1(d) in the main text]. Theagreement between theory and experiment is good. run length distribution using our theory [Eq.(S3)] along with the 4 constraints describedabove are used to determine the model parameters at zero load, k , k +0 , k − , γ , and K T .Subsequently, we used the data for average velocity vs load at different ATP concentrationsin Ref.[11] to obtain d + and d − . The best fit parameters are listed in Table.1 and Table.2in the main text for the 2HB waiting and 1HB waiting model, respectively.22
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