Fitting Force-dependent Kinetic Models to Single-Molecule Dwell Time Records

Transcription

1 Fitting Force-dependent Kinetic Models to Single-Molecule Dwell Time Records Abstract Micromanipulation techniques such as optical tweezers atomic force microscopy allow us to identify characterize force-dependent transitions in molecular machines mechanochemical cycles by studying how dwelltime distributions vary as assisting or hindering loads are applied. We present an new simple maximum likelihood method for estimating the zero-force rate distance to the transition state. As a practical benefit to the experimenter, our method allows for relaxed experimental conditions; one no longer has to perform dwell time measurements repeatedly at very precisely controlled fixed forces, but can use data obtained at widely varying forces as long as the forces are measured or known. Moreover, our method is generalizable to more complicated kinetic pathways including those in which some rates do not vary with force. Introduction The development of micromanipulation techniques that allow measurement control of piconewton-scale forces nanometer-scale displacements with time resolution on the order of milliseconds, such as optical tweezers or atomic force microscopy, has made it possible to probe, under mechanical load, the kinetics of biomolecular processes at the single-molecule level. Such high-resolution measurements in combination with novel analysis methods have enabled determination of the distributions of durations of experimentally observable states within the underlying mechanochemical pathway,,3. Such experiments have proved invaluable in the study of biomolecular machines, including the processive motion of cargo-transporting motor proteins such as kinesin myosin-v, enzyme-substrate interactions involving mechanical deformations, such as initiation by T7 RNA polymerase or DNA cleavage by restriction enzymes. 4,5,6 * " ' 4 * 3 * + %,! & / " ', + " # % " # -. Figure : Illustration of the experimental setup for kinesin attached to a bead, translocating along a microtubule subject to optical of the experimental setup for kinesin attached to a bead, translocating tweezersfigure pulling : on Illustration bead a microtubule microscope slide coverslip, subject tweezers is pulling The along distribution of dwellbound timestoinathe experimentally observable statestoofoptical such systems determined by the underlying kinetic pathway. Force dependence of the rate of a single mechanochemical sub-step in a pathway may backusing on bead. be modeled transition-state theory, in which the rate Gthirteen microtubule s structure consists of a sheet of exp typically or fourteen different parallel k. kb T tubulin protofilaments wrapped to form a cylinder, kinesin remains parallel to individual protofilaments so that it effectively undergoes one-dimensional motion Ray et al., 993. Securing individual microtubules to the surface of a microscope coverslip in a predominantly uniform direction, a biophysicist may record the linear displacement of a bead pulled by a

2 Statistics Case Studies: Single Molecule Dwell Times Math 363 Spring 9 Here k B is Boltzmann s constant T is the absolute temperature. Taking displacement to be our reaction coordinate, the free energy G may be split into the sum of the work done against or assisted by the external force the unloaded free energy, G ± F δ where δ is the distance to the transition state the sign depends on whether the applied force helps or hinders the reaction. We thus have k = k e ±F δ/k BT, where k is the zero-force rate, for the form of the force-dependent rates in the system. Current Method Estimation of rates by fitting force-dependent dwell-time distributions to this model has to date been predominantly done by an ad hoc procedure, wherein data are grouped into force bins, lifetime distributions are fit to each bin, the resulting force-dependent rates are fit to???. This procedure only works well when the underlying kinetics consist of a single rate-limiting step force bins are kept narrow by taking many measurements are taken at each of a series of fixed forces. Problems arise when the data cannot, due to the nature of the experiment, be collected at precisely predetermined forces???. Because the dwell times vary nonlinearly with force, the mean dwell time computed from data taken at a range of forces is not an accurate estimator of the mean dwell time at the force at a force-bin s center. We illustrate this in Figure??, which shows the distribution of mean dwell times estimated using a force bin one piconewton in width containing simulated force-dwell time pairs. To compute each mean dwell time, we drew force values from the uniform distribution from 3.5 pn to 4.5 pn, assigned to each force a rom, exponentially-distributed dwell time drawn from a distribution with density ft τ, δ, F = τ ef δ/k BT exp t τ e F δ/k BT with k B T =4. pn nm, δ=nm τ =3 arbitrary time units. The mean dwell time for the bin was then computed by averaging these rom dwell times. This was repeated ten thous times to generate the distribution shown in the figure. It is clear that the mean dwell time is biased towards higher values than the mean expected at the bin s center. This bias does not diminish if we increase the number of points in a bin even to experimentally unobtainable numbers. Moreover, the distribution of such an estimate with points in the bin is non-normal. Similar bias non-normality not shown were found when the forces of a force-bin were drawn from normal distributions. In both cases, the bias gradually decreased as the width of the uniform or normal distribution was lessened. Such narrow force binning is not always experimentally possible, especially when high-force events are intrinsically rare or difficult to measure. 3 Fitting a Single, Force-Dependent Transition Rather than grouping data into force bins, we propose fitting force-dependent kinetic models to data taken at all forces by maximum likelihood. For most real-world probability distributions, maximum likelihood estimators are guaranteed to be consistent, which is to say that they converge to the correct result as the number of samples increases?. Such a method also allows for the fitting of models in which some transitions are force-dependent others are not, removes the need for binning. In the case of a single, force-dependent transition, such an approach yields an elegant graphical method of estimating the mean zero-force lifetime τ or transition rate k distance to the transition state δ. Consider the simple reaction, bound unbound, describing an irreversible detachment of two molecules, e.g. a single myosin head head an actin filament microtubule as in the experiment of Nishizaka et al.?. Since there are no substeps, dwell times in the bound state are

3 Statistics Case Studies: Single Molecule Dwell Times Math 363 Spring 9 5 A 5 Count Estimated τ Figure : Histogram of mean lifetimes inferred from, data sets simulated as described in the text. These sets were then split into force bins. Histograms A, B, C corresponding to fits to,, 5 force bins, respectively. Mean dwell times Figureforces : were Histogram computed forof eachunloaded bin, ˆτ bin was zero-force then fit to these means using lifetimes the stard weighted inferred leastusing squares method. the ad Dashed hoc bin-fitlinemethod indicates expected frommean, lifetimedata force-bin sets center. simulated Note that the estimated as follows: value of τ, stronglyforces depends on were the bindrawn size from that the optimally sized bins cannot be known in advance. U pn, pn. Exponentially distributed rom lifetimes were then computed for each force using drawn from thea single- Arrhenius-like exponential distribution. rate law Allowing Eq, thek transition B T =4. to bepn force dependent nm, δ = as above, nm, the observed τ = k = dwell time given a measured applied force is drawn from the probability distribution having density 3 arbitrary The log-likelihood units. These functionsets for the were experiment then with split independently into force measured bins. forces Histograms F = F, FA,,.. B,., F N C corresponding corresponding to fits independent to, dwell, times t 5 = t force, t,.. bins,., c N isrespectively. Mean lifetimes forces were computed for each bin, τf =τ e F δ/kbt was then fit to these means using the stard weighted least squares method. log Lτ, δ t, Dashed F = n log vertical τ + δ N n lines indicate thet i e Fiδ/k BT correct value of τ. 3 Maximizing the log likelihood requires that both k B T δ log Lτ, δ t, F = τ Nτ τ log Lτ, δ t, F = F Nτ F i τ B C N t i e Fiδ/kBT = N t i F i e Fiδ/kBT = where F is the mean value of the forces. These equations do not separate to produce closed forms for the maximum likelhood estimators ˆδ ˆτ. However, they can be rearranged to obtain ˆτ = N t i e Fi ˆδ/k B T 4 N ˆτ = N F N t i F i e Fi ˆδ/k B T. 5 This facilitates a simple graphical interpretation for ˆτ ˆδ as the intersections of these two functions of ˆδ. 3

4 Statistics Case Studies: Single Molecule Dwell Times Math 363 Spring " Figure 3: Example of the maximum likelihood estimator using values described in the text. This is found by the intersection of 4 5 in the ˆδ-ˆτ plane. The direction of the axes of the ellipse are derived from the covariance of δ τ as given by the off diagonal elements in the inverse of the Fisher information matrix 6. igure 3: Example of the model-fitting technique presented in Section. The intersection of the In addition to providing estimates ˆτ urves given in Equations 3 4, marked ˆδ, the theory of maximum likelihood estimation also provides stard errors of these estimates in a straightforward manner. with Thea covariance, gives matrix of the asymptotic maximum normallikelihood distribution of estimators these estimators has a simple closed form. The expected Fisher information is given by f the unloaded mean time τ the distance to transition state δ. The ellipse demarcates he region of 95% confidence, the true value /τ Iτ, δ = N of F δ/τ k B T τ is marked with a +. Fit was erformed on, simulated data, with forces F /τ k B T F /k B T between zero. ten piconewtons drawn from Its inverse, the covariance matrix, is uniform distribution dwell times drawn from the force-dependent distribution function iven in Eq., δ= nm, τ =3 arbitrary Iτ, δ units, k τ = l F τ B T =4. k B T pn nm. F. 6 NvarF! τ k B T F k B T Iτ, δ Iτ, δ are the respective stard errors of ˆτ ˆδ. Plotted in Figure?? is an example of the result of this procedure. One thous pairs of independent measurements were simulated by drawing a force uniformly distributed between pn. then drawing the corresponding lifetime from the exponential distribution given in Eq., As above, δ= nm, k B =4. pn nm, τ =3 arbitrary units. The true values of δ τ are marked with a +. The intersection, located numerically marked with a, of the curves given in Equations 4 5 plotted gives the maximum likelihood estimators ˆτ =.99 ±.9, ˆδ =.4 ±.45. The region of 95% confidence is demarcated by the ellipse. 4 Discussion The procedure we present in Section is straightforward computationally rapid. It is easier to apply than the ad hoc method in common use. Moreover, for this ad hoc method, the mean force is highly dependent on the choice of bin size consequently is unreliable as an estimation technique. Locating the intersection of two curves is simpler than the force-binning, fitting single-force likelihoods, then fitting curves to parameters. Likelihood functions for even relatively simple models can become rather complicated. Solutions like 4 5 can sometimes be found. For more complex models, the likelihood given or its logarithm may be maximized using an entirely numerical procedure. In addition to avoiding complications surrounding binning, maximum likelihood estimation of kinetic model parameters allows us to make the most of available data, as estimation is done using all of the information collected in the experiment, at once. One no longer has to perform dwell time measurements repeatedly at very precisely controlled fixed forces, but can use data obtained at widely varying forces as long as the forces are measured or known. In the 3

5 Statistics Case Studies: Single Molecule Dwell Times Math 363 Spring 9 Count 5 5 A 5 5 B Count C D ˆτ ˆδ Figure 4: Histograms of ˆτ A,C ˆδ B,D estimated by finding the intersection of the curves given in Eqs A B were Figure constructed 4: Histograms from, sets of of ˆτ A,C force-lifetime ˆδ pairs B,D generated estimated as for Figure by finding??; C the D were intersection constuctedof fromthe, sets curves of, given such pairs. in Eqs. Dashed 3 vertical 4. lines A indicate B were the true constructed values of τ from δ. Compare,tosets Figure of. force-lifetime pairs generated as for Figure 3; C D were constucted from, sets of, such pairs. simple Dashed one-step vertical caselines treatedindicate Section the, true we see values from the of τcovariance δ. Compare matrix Eq. to 6, Figure which scales. as varf, that we benefit from taking measurements over a wide range of forces, incurring a penalty in estimating the unloaded rate as the mean square force becomes high. We expect the same qualitative behavior a benefit for taking measurements at a broad range of forces but a penalty for speeding up the reaction with high assisting forces when treating more complicated models. Estimating the parameters of force-dependent models by maximum likelihood opens the possibility of using model selection techniques such as information criterion or likelihood ratio tests to complement biochemical biophysical experiments. References [] L. S. Milescu, A. Yildiz, P. R. Selvin, F. Sachs, 6 Biophysical Journal 9, [] B. C. Carter, M. Vershinin, S. P. Gross, 8 Biophysical Journal 94, [3] S. N. Block 7 Biophysical Journal 9, [4] A. Mehta Journal of Cel l Science 4, [5] J.-F. Allem, D. Bensimon, V. Croquette 3 Current Opinion in Structural Biology 3, [6] G. M. Skinner, C. G. Baumann, D. M. Quinn, J. E. Molloy, J. G. Hoggett 4 Journal of Biological Chemistry 79,

6 Statistics Case Studies: Single Molecule Dwell Times Math 363 Spring 9 5 Model STATISTICAL PROCEDURES Let t i, i n be the observed values of n independent exponential rom variables with parameter Here, the model parameters, exp F iδ τ k B T. τ is the characteristic disassociation time with no force, δ is the displacement from equilbrium. In addiition, F i is the force associated to the i-th observation, k B is Boltzmann s constant, T is the absolute temperature Thus, an observed time t based on a force F is drawn from a probability distribution having density ft τ, δ, F = τ ef δ/ exp t τ e F δ/, t In particular, the mean E τ,δt = τ e F δ/k BT. 6 Likelihood Function The likelihood function for the experiment The score function has components Lτ, δ t, F = n log Lτ, δ t, F = n log τ + δ τ log Lτ, δ t, F = n δ log Lτ, δ t, F = = n τ efiδ/ exp t i τ e Fiδ/. 5 F i τ τ nτ F i τ F nτ n t i e Fiδ/. t i e Fiδ/. t i F i e Fiδ/ t i F i e Fiδ/

7 Statistics Case Studies: Single Molecule Dwell Times Math 363 Spring 9 Consequently, the maximum likelihood estimates ˆτ, ˆδ satisfy = ˆτ nˆτ = F nˆτ t i e Fiδ/, or ˆτ = n Fi ˆδ/ t i F i e or ˆτ = n F t i e Fi ˆδ/, t i F i e Fi ˆδ/. Thus, to find the maximum likelihood estimates, graph the values of τ in each of the two relations above as a function of δ. The intersection of these two graphs will give the maximum likelihood estimates. 7 Fisher Information Matrix Write θ = τ, δ let ˆθ n be the maximum likelihood estimator based on n observations. We shall see that in these circumstances that ˆθ n is consistent, that is, if θ is the state of nature, In addition, ˆθ n θ in probability as n. nˆθn θ W in distribution as n where W is a mean zero normal rom vector with covariance matrix the inverse of the Fisher information matrix Iθ. Consequently, the maximum likelihood estimator is asymptotically efficient in the sense that the ratio of any other consistent estimator with the variance of this estimator is at most. To find the Fisher information matrix, note that Next, τ I τ, δ = E τ,δ = n τ log Lτ, δ t, F = n [ τ nτ 3 τ nτ 3 ] log Lτ, δ t, F = n τ t i e Fiδ/. nτ 3 e Fiδ/ E τ,δt i e Fiδ/ τ e Fiδ/ = n δ log Lτ, δ t, F = n nτ t i F i F i efiδ/ [ ] I τ, δ = E τ,δ δ log Lτ, δ t, F = Finally, for the mutual information, F i = n F δτ log Lτ, δ t, F = τ 6 = τ τ nτ = τ = n τ. t i Fi e Fiδ/ Fi e Fiδ/ E τ,δt i t i F i e Fiδ/

8 Statistics Case Studies: Single Molecule Dwell Times Math 363 Spring 9 [ ] I τ, δ = E τ,δ log Lτ, δ t, F δτ = τ F i = n F τ = τ F i e Fiδ/ E τ,δt i Thus, the information matrix /τ F /τ Iτ, δ = n F /τ F / The deteminant is varf /τ the inverse Iτ, δ = τ F / F /τ = τ F τ F nvarf F /τ /τ nvarf τ F Now, we can use stard z-statistics to perform interval estimation hypothesis testing.. 7