Conformational dynamics through an intermediate Ashok Garai, Yaojun Zhang, and Olga K. Dudko Citation: The Journal of Chemical Physics 140, 135101 (2014); doi: 10.1063/1.4869869 View online: http://dx.doi.org/10.1063/1.4869869 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Modulating material interfaces through biologically-inspired intermediates Appl. Phys. Lett. 99, 233701 (2011); 10.1063/1.3651756 Effects of osmolytes on the helical conformation of model peptide: Molecular dynamics simulation J. Chem. Phys. 134, 035104 (2011); 10.1063/1.3530072 Low molecular weight oligomers of amyloid peptides display β -barrel conformations: A replica exchange molecular dynamics study in explicit solvent J. Chem. Phys. 132, 165103 (2010); 10.1063/1.3385470 Role of conformational dynamics in kinetics of an enzymatic cycle in a nonequilibrium steady state J. Chem. Phys. 131, 065104 (2009); 10.1063/1.3207274 Solvent effects on conformational dynamics of proteins: Cytochrome c in a dried trehalose film J. Chem. Phys. 117, 4594 (2002); 10.1063/1.1498459
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THE JOURNAL OF CHEMICAL PHYSICS 140, 135101 (2014)
Conformational dynamics through an intermediate Ashok Garai,a) Yaojun Zhang,a) and Olga K. Dudkob) Department of Physics and Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California 92093, USA
(Received 5 February 2014; accepted 17 March 2014; published online 2 April 2014) The self-assembly of biological and synthetic nanostructures commonly proceeds via intermediate states. In living systems in particular, the intermediates have the capacity to tilt the balance between functional and potentially fatal behavior. This work develops a statistical mechanical treatment of conformational dynamics through an intermediate under a variable force. An analytical solution is derived for the key experimentally measurable quantity—the distribution of forces at which a conformational transition occurs. The solution reveals rich kinetics over a broad range of parameters and enables one to locate the intermediate and extract the activation barriers and rate constants. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4869869] A conformational change is the basis of phenomena ranging from chemical synthesis of nanostructures1 to the selfassembly of the machinery of the living cell.2 Conformational changes usually involve intermediates—regions of configurational space where the system dwells on the way from the initial to the final state. Because of their transient nature, intermediates are challenging to detect, and all the more so to characterize. Yet, their characterization is vital: not only are the intermediates the steps leading to the functional state, but they are also the vulnerable steps as they can undergo larger thermal fluctuations and are thus prone to misfolding and aggregation that may prove fatal to a living system.3 Intermediates involved in conformational changes can be probed directly, using single-molecule force techniques.4–7 When the force applied to a macromolecule is kept constant, “hopping” in and out of an intermediate can be observed. Such force clamp measurements report force-dependent transition rates directly, although typically in a limited force range. When the applied force is continuously varied, intermediates can be detected in the force-extension curves. Such force ramp measurements report the distribution P(F) of transition forces, which is a treasury of mechanistic information, but requires a theoretical foundation for this information to be extracted. When attempting to develop such a theoretical foundation, the first fact to face is that intermediates complicate the problem simply because they entail new types of transitions. Add to this a varying force, and the rates of all transitions become time-dependent. These complications, even in the presence of a single intermediate, give rise to interesting kinetic scenarios. One notable scenario8 occurs when the major energy barrier disappears under force and an initially minor barrier emerges to dominate the kinetics, all within a single measurement. Conformational transitions via a single intermediate, beside being ubiquitous in biology and nonbiological applications,1, 9 also provide insights into more complex a) A. Garai and Y. Zhang contributed equally to this work. b) Author to whom correspondence should be addressed. Electronic mail:
[email protected]. 0021-9606/2014/140(13)/135101/5/$30.00
dynamics. In this article, we show that the kinetics of a system with an intermediate, subject to a steadily increasing force [Fig. 1(a)], can be described analytically with a microscopic theory of a substantial generality. The theory culminates in an analytical expression for the transition force distribution P(F), an experimentally measurable quantity. The expression predicts the spectrum of behaviors that P(F) exhibits across a broad range of parameters, and reveals the emergence of distinct features [Fig. 1(b)]. These features provide a route to identifying the mechanism of a conformational transition among multiple candidate mechanisms. Through the established analytical relationship between the observable features and the microscopic parameters, the derived expression enables the extraction of these parameters (Fig. 2) from experimental data. By definition, an intermediate (I) is a metastable state separated from the native (N) and unfolded (U) states by significant energy barriers [Fig. 1(a)]. Consider an experiment in which a steadily increasing force is applied to a system with an intermediate, and the value of the force needed to transit into U is recorded. This transition force is determined by the parameters of the energy landscape, the force loading rate, and thermal noise. Due to the stochastic nature of the transition, the transition forces recorded in multiple repeats of this experiment are distributed according to a certain probability density P(F). Conformational changes through an intermediate under a force F(t) are governed by the master equation for the survival probabilities Si (t) in states i = {N, I, U}: ⎡˙ ⎤ ⎡ ⎤⎡ ⎤ SN (t) kI N (t) 0 SN (t) −kNI (t) ⎢ ˙ ⎥ ⎢ ⎥⎢ ⎥ ⎣ SI (t) ⎦=⎣ kNI (t) −kI N (t) − kI U (t) 0 ⎦⎣ SI (t) ⎦ S˙U (t)
0
kI U (t)
0
SU (t) (1)
While a single-intermediate system with constant rate coefficients kij is a textbook example of an analytically tractable kinetic description,9 the system in Eq. (1) is not: a non-trivial aspect of its kinetics is the time-varying perturbation due to the force F(t). The force introduces the time-dependence
140, 135101-1
© 2014 AIP Publishing LLC
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J. Chem. Phys. 140, 135101 (2014)
To convert Eq. (2) into a practical, closed-form expression, we must (i) determine the functional form of the individual force-dependent rates kij (F(t)), (ii) integrate these rates to find gij (F(t)), and (iii) perform the integration in Eq. (2). Step (i) is accomplished by applying Kramers formalism, which treats an elementary reaction as Brownian motion,10 to free energy profiles of different shapes captured by a scaling factor ν, and further noting that the pulling device is typically soft compared to the N and I states,11 κ 2kB T/x‡2 : ‡
FIG. 1. (a) Free energy profile with the native (N), intermediate (I), and unfolded/unbound (U) states, perturbed by a force F(t). (b) Transitions into U result in a distribution of transition forces, P(F), measurable in experiment. The analytical theory reveals how a bimodal (upper) or a unimodal (lower) P(F) emerge from the interplay between the population in I and the rate at which U receives population from I.
in the rate coefficients kij (t), thereby not only enriching the behavior of the probabilities Si (t)—but also making it challenging to solve Eq. (1) analytically. Nevertheless, we will now show that this system is amenable to a fairly general analytical treatment. The quantity of interest, the experimentally measurable distribution of forces at which the system transits into the final state (U), is defined as P (F ) ≡ kI U (F )SI (F )/F˙ , where F˙ = κV is the force loading rate, κ is the spring constant of the pulling device, and V the pulling speed. Because the external force lowers the barriers for forward transitions and raises the barriers for backward transitions [Fig. 1(a)], beyond very small forces, transitions I → N become rare: kIN ≈ 0. Expressing SI (F) and SN (F) as the respective formal solutions of the second and first equations in Eq. (1), we obtain the integral expression for the quantity of interest: kI U (F ) −gI U (F ) F gI U (f ) kNI (f ) −gNI (f ) e e P (F ) = e df, (2) F˙ f˙ 0 F where gij (F ) ≡ 0 kij (f )df/f˙ and the change of the variable of integration from time t to force F is straightforward due to a monotonic increase of F(t) prior to a transition. Note that a transition via an intermediate, unlike its simple single-barrier counterpart, involves a generally non-negligible drop in the force from f to f occurring in the course of the transition—upon escape from N to I—due to partial unfolding; Eq. (2) accounts for this drop.
‡
FIG. 2. Intrinsic parameters sought to be reconstructed: the heights Gij of the activation barriers, the locations diate, and kinetic rates kij0 .
‡ xij
of these barriers and of the interme-
⎡
Gij
⎣ ‡ 1 νF xij ν −1 kB T 1− 0 e kij (F ) = kij 1 − ‡ Gij
1−
‡ νF xij ‡ Gij
ν1 ⎤ ⎦
,
(3)
where kB is the Boltzmann constant and T the temperature. The integral of Eq. (3) with respect to force can be solved exactly, which readily takes us through step (ii): ‡
⎡
ij ⎣ G 1− kB T kB T kij0 e gij (F ) = ‡ F˙ xij
1−
‡ νF xij ‡ Gij
ν1 ⎤ ⎦
−1 .
(4)
To fulfill the final step (iii), we note a difference in behavior with respect to force of the two functions: a gradually decreasing e−gNI (F ) and a steeply increasing kI U (F )egI U (F ) (a special case where this behavior does not hold is analyzed separately). Then, the steepest descent method of evaluating the integral in Eq. (2) yields (see the supplementary material12 ) the sought-for analytical solution: 1 kI U (F )e−gI U (F ) [1 − e−gNI (F ) ] + kNI (F ) P (F ) = F˙ × e−gNI (F ) [1 − e−gI U (F ) (1 + gI U (F ))] 1 + gI−1 U (F ) (5)
with kij and gij given by Eqs. (3) and (4) and F = F − F accounting for the force drop F upon transition N → I, discussed above. The analytical expression in Eq. (5) for the experimentally measurable distribution of transition forces in a system with an intermediate is the key result of this paper. The predictive power of Eq. (5) is explored in Fig. 3 through a systematic overview of representative free energy landscape topologies possessing an intermediate. Equation (5) is applicable to topologies I, II, III, and V; a discussion of topology IV will follow later. The transition state (‡) is defined as the major barrier in the absence of force. The resulting kinetic scenarios under a force-ramp can be understood in terms of two key properties of the intermediate I. First, the location of I relative to ‡: before (“early intermediate”) or after (“late intermediate”). Second, the pliability of I relative to that ‡ ‡ of the native state N: a soft I implies xI U > xNI , a stiff I implies the inverse. For an early and stiff intermediate (case II in Fig. 3) and for a late and soft intermediate (case III), the kinetics at all forces are dominated by the transition state. However, a switch in dominance from the transition state to the initially minor barrier occurs in two topologies: an early and soft intermediate (case I) and a late and stiff intermediate (case IV). When two comparable barriers are separated by an intermediate of a pliability similar to that of state N (case V), both barriers equally dominate the kinetics at all forces.
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J. Chem. Phys. 140, 135101 (2014)
FIG. 3. A systematic overview of force-induced transitions through an intermediate. (a) Representative topologies (I-V) of the free energy profile with an intermediate at three values of force. (b) Representative force-extension curves at low, medium, and high force loading rate κV . Arrows indicate the measured rupture forces. (c) Rupture force distributions P(F) for the transition into the final state (U) at three values of κV . Histograms are simulated data, lines are a global fit to the analytical theory (see the supplementary material12 for details on cases III and IV). (d) Most probable transition force Fpeak vs. loading rate (the force spectrum) for two types of transitions: I → U (triangles, these are the peaks of the histograms in (c)) and N → I (circles). Black/gray lines are the peak peak predictions of Eq. (8), F peak = FI U (κV ) and F peak = FNI (κV ) − F , respectively, if the outer/inner barrier were to dominate at all κV . Note that the triangles follow the black/gray line whenever the outer/inner barrier is dominating. The triangles switch between the black and gray lines in cases I and IV where the switch in the dominant barriers occurs. (e) Individual force-dependent rates, Eq. (3), for transitions N → I, I → U, and I → N, and the effective rate NI (F )kI U (F ) keff (F ) = kNI (Fk)+k . Beyond very low forces, the effective rate follows the rate limited by the dominant barrier. I N (F )+kI U (F )
Equation (5) further reveals that the presence of an intermediate may make the transition force distribution undergo a transformation from unimodal to bimodal and back to unimodal when increasing the loading rate (Fig. 3(c), case I). The possibility of a bimodal force distribution due to an intermediate has been pointed out by Strunz et al.13 who simulated the master equation, Eq. (1), numerically. We now show how the bimodality in the force distribution emerges, as a transient scenario, from the general analytical solution [Eq. (5)] to the master equation. What is the origin of the bimodality in the transition force distribution? Equation (5) shows that the condition for bimodality is an early and soft intermediate (case I in Fig. 3). At short times, and hence low forces, population accumulates in I, blocked from entering U by a high transition state (Fig. 4). However, because the transition state is soft and thus lowered rapidly, I soon begins to release this accumulated population (hence the increase in the measured P(F)), while the rest of the population remains trapped in the stiff N. Once
the population in I is depleted, a drop in P(F) occurs, completing the first peak. This short-time behavior is captured by the term kI U (F )e−gI U (F ) in Eq. (5). At long times, and hence high forces, the stiff inner barrier is lowered sufficiently for N to release its trapped population, leading to a new rise in the measured P(F). Once the remaining population is released from N, P(F) drops again, completing the second peak. This long-time behavior is captured by the term kNI (F )e−gNI (F ) in Eq. (5). Remarkably, Eq. (5) reveals that the bimodality caused by the presence of an intermediate is always described by a particular configuration of the two peaks, namely a narrow peak at lower forces followed by a broader peak at higher forces (Fig. 3(c), case I). This distinct, and persistent, configuration of the two peaks is a consequence of the soft nature of I relative to that of N, a condition necessary for the bimodality to emerge. Indeed, a transition from a softer state (here, I) will lead to a more narrow distribution (here, first peak) of transition forces, because a soft state rapidly releases its population,
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135101-4
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J. Chem. Phys. 140, 135101 (2014) peak where F = F + F − F˙ τ with τ ≡ kI−1 U (FNI (κV ) − F ) accounts for a drop in the force from F in state N to (F − F˙ τ ) upon escape into I, and a subsequent increase to the force F (the measured value) upon escape into U. peak The value of FNI (κV ) is given by Eq. (8) with α = 1 and ij = NI. The most probable force, Fpeak , in the force distribution P(F) for the transition I → U is described by (see the supplementary material12 ) ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ peak ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎩
Fij
dip med
⎪ ⎪ ⎪ ⎪ ⎭
‡
(κV ) =
Gij ‡
νxij
ν ‡ ακV xij ln 1− 1− . ‡ kB T kij0 Gij
kB T
(8)
FIG. 4. The origin of bimodality in the transition force distribution of a system with an intermediate. The distribution, P(F), is equivalent to the flux into the final state (U), given by the rate (set by the barrier) times the population. The arrow widths reflect the fluxes at different instants.
giving less time for thermal fluctuations to broaden the resulting force distribution. The described distinct features of the bimodality in the force distribution can be used as a basis for identifying or eliminating the intermediate as the mechanism of a conformational transition among several candidate mechanisms. For topology IV, the expressions for P(F), derived in the supplementary material12 , are as follows. At high loading rates, where the most probable forces for transitions N → I and I → U begin to diverge (Fig. 3(d), case IV), P (F )
1 gI U (F med ) e [1 − e−gNI (F ) ]kI U (F )e−gI U (F ) , F˙
(6)
med (κV ) − F with where F = F + F and F med = FNI med FNI (κV ) being the median force for transition N → I [given by Eq. (8) with α = ln 2 and ij = NI]. At low loading rates (but high enough for transitions I → N to be negligible), where the most probable forces for transitions N → I and I → U are still close,
P (F )
1 kNI (F )e−gNI (F ) , ˙ F
(7)
When the force distribution for I → U is bimodal, the locations of the first and second peaks can be found from peak peak Eq. (8) as F peak1 = FI U (κV ) and F peak2 = FNI (κV ) − F with α = 1. We further find (see supporting material12 ) that Eq. (8) also captures the location of the dip in the dip bimodal distribution as F dip = FI U (κV ) with α = 8. When the force distribution for I → U is unimodal, the location of its peak can be found from Eq. (8) as F peak peak = FNI (κV ) − F if the first barrier is dominating and as peak F peak = FI U (κV ) if the second barrier is dominating, with α = 1. Thus, Eq. (8) unifies, via α, a set of relationships between observable features (the peaks and the dip in the force distribution) and the parameters of the system (Fig. 2) and, as such, provides additional means for extracting these parameters. The performance of the developed theory as a fitting tool is reported in Fig. 3 and Table I with data from Brownian dynamics simulations of systems with an early and soft intermediate (case I in Fig. 3) at parameters representative of the optical tweezers and AFM setups. Table SI reports the fitting results for all other topologies (cases II-V in Fig. 3) (see the supplementary material12 ). In principle, a fit of the force histograms for the final transition (I → U) to Eq. (5) in a broad range of loading rates recovers the heights and locations of the two barriers and the corresponding transition rates. In practice, however, an insufficient quality of the histograms may affect the robustness of such a six-parameter fit, and the following two-step procedure may prove more efficient. First, force histograms for the first transition, N → I, at different values of κV are fitted globally (i.e., with a unique set of three fitting parameters for all histograms) to the singlebarrier expression pNI (F ) = F1˙ kNI (F )e−gNI (F )11 with kNI (F) and gNI (F) in Eqs. (3) and (4) at a fixed ν (0.5 in Fig. 3). This
TABLE I. Kinetic and energetic parameters of a system with an intermediate (Fig. 2), extracted from the fit of the simulated data to the derived analytical theory. The values of the stiffness κ (in pN/nm) represent optical tweezers and an AFM. ‡
True, κ= 0.1 Fit True, κ= 5 Fit
‡
‡
‡
‡
‡
xNI [nm]
GNI [kB T]
0 ln(kNI [s−1 ])
xI N [nm]
GI N [kB T]
ln(kI0N [s−1 ])
xI U [nm]
GI U [kB T]
ln(kI0U [s−1 ])
1.00 1.08 ± 0.02 0.30 0.31 ± 0.01
16.0 15.8 ± 0.9 18.0 18.7 ± 2.6
− 4.5 − 4.6 ± 0.1 − 4.4 − 3.9 ± 0.1
1.80 1.72 ± 0.02 0.60 0.59 ± 0.01
8.0 9.7 ± 0.8 8.0 11.6 ± 2.8
2.2 1.5 ± 0.1 4.2 3.2 ± 0.1
4.5 4.5 ± 0.6 1.5 1.8 ± 0.3
22.0 26.4 ± 3.5 25.0 26.6 ± 0.9
− 13.2 − 13.5 ± 1.1 − 15.2 − 16.3 ± 1.6
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135101-5
Garai, Zhang, and Dudko ‡
J. Chem. Phys. 140, 135101 (2014) ‡
fit yields the height GNI and location xNI of the first barrier 0 . Next, force histograms and the associated intrinsic rate kNI for the second transition, I → U, are globally fitted to Eq. (5) ‡ ‡ 0 fixed at the values extracted from with GNI , xNI , and kNI the fit of the first transition. The average drop in the force, F, is read off the force-extension curves (see the supplementary ‡ ‡ material12 ). This fit yields the height GI U and location xI U 0 of the second barrier relative to I and the intrinsic rate kI U . To determine whether Eq. (5) is applicable to the system under study, we locate the peaks in the force histograms for N → I and I → U (circles and triangles, respectively, in Fig. 3(d)) to determine the most probable forces for these transitions, and plot these forces together as a function of the loading rate κV . If these forces are close at low κV and diverge with increasing κV , which matches the pattern of the force spectrum for case IV in Fig. 3(d), then Eq. (6) can be used to fit the force histograms for I → U at high κV and Eq. (7) at low κV . In all other cases, Eq. (5) can be used to fit the force histograms for I → U in the entire κV range. The remaining parameters—the location of the intermediate I, the intrinsic rate for transition I → N and the activation barrier separating state I from state N (Fig. 2)—can be estimated as follows (see the supplementary material12 ): ‡
‡
‡
xI N ≈ F /κeff − xNI , kI0N ≈ kI N (F1/2 )eF1/2 xI N /kB T , (9) 0 ‡ ‡ − ln kI0N , GI N ≈ GNI + kB T ln kNI where the mid-point force F1/2 and kIN (F1/2 ) can be determined from a force-clamp measurement and κ eff is the effective spring constant of the pulling device and the linker (if present) tethering the molecule to the device. The analytical framework developed above should be used with caution at very high and very low forces. At very high forces, a limited time resolution may prevent the detection of the intermediate in the force extension curves; if in-
creasing the time resolution is not possible, these data sets should not be included in the fit. At very low forces, the assumptions of the theory may no longer be justified if backward transitions I → N become non-negligible or if the drop in the force (F) upon transition N → I varies significantly from its average. The developed theory unifies a broad range of macromolecular systems and experimental techniques. The established general principles provide a quantitative framework for mechanistic understanding of multistep conformational transitions and may point to universal strategies for influencing specific steps in the related processes. This research was supported by NSF CAREER Grant No. MCB-0845099 and NSF CTBP Grant No. PHY-0822283. 1 G.
M. Whitesides, J. P. Mathias, and C. T. Seto, Science 254, 1312 (1991). 2 B. Alberts et al., Molecular Biology of the Cell (Garland Science, New York, 2008). 3 M. Vendruscolo and C. M. Dobson, Nat. Chem. Biol. 9, 216 (2013). 4 P. E. Marszalek, H. Lu, H. Li, M. Carrion-Vazquez, A. F. Oberhauser, K. Schulten, and Julio M. Fernandez, Nature (London) 402, 100 (1999). 5 C. Cecconi, E. A. Shank, C. Bustamante, and S. Marqusee, Science 309, 2057 (2005). 6 M. T. Woodside, P. C. Anthony, W. M. Behnke-Parks, K. Larizadeh, D. Herschlag, and S. M. Block, Science 314, 1001 (2006). 7 J. Stigler, F. Ziegler, A. Gieseke, J. C. M. Gebhardt, and M. Rief, Science 334, 512 (2011). 8 R. Merkel, P. Nassoy, A. Leung, K. Ritchie, and E. Evans, Nature (London) 397, 50 (1999). 9 A. R. Fersht, Structure and Mechanism in Protein Science (W. H. Freeman, New York, 1999). 10 H. Kramers, Physica 7, 284 (1940). 11 O. K. Dudko, G. Hummer, and A. Szabo, Phys. Rev. Lett. 96, 108101 (2006). 12 See supplementary material at http://dx.doi.org/10.1063/1.4869869 for details on the derivations and simulations. 13 T. Strunz, K. Oroszlan, I. Schumakovitch, H.-J. Güntherodt, and M. Hegner, Biophys. J. 79, 1206 (2000).
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THE JOURNAL OF CHEMICAL PHYSICS 140, 135101 (2014)
Conformational dynamics through an intermediate Ashok Garai,a) Yaojun Zhang,a) and Olga K. Dudkob) Department of Physics and Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California 92093, USA
(Received 5 February 2014; accepted 17 March 2014; published online 2 April 2014) The self-assembly of biological and synthetic nanostructures commonly proceeds via intermediate states. In living systems in particular, the intermediates have the capacity to tilt the balance between functional and potentially fatal behavior. This work develops a statistical mechanical treatment of conformational dynamics through an intermediate under a variable force. An analytical solution is derived for the key experimentally measurable quantity—the distribution of forces at which a conformational transition occurs. The solution reveals rich kinetics over a broad range of parameters and enables one to locate the intermediate and extract the activation barriers and rate constants. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4869869] A conformational change is the basis of phenomena ranging from chemical synthesis of nanostructures1 to the selfassembly of the machinery of the living cell.2 Conformational changes usually involve intermediates—regions of configurational space where the system dwells on the way from the initial to the final state. Because of their transient nature, intermediates are challenging to detect, and all the more so to characterize. Yet, their characterization is vital: not only are the intermediates the steps leading to the functional state, but they are also the vulnerable steps as they can undergo larger thermal fluctuations and are thus prone to misfolding and aggregation that may prove fatal to a living system.3 Intermediates involved in conformational changes can be probed directly, using single-molecule force techniques.4–7 When the force applied to a macromolecule is kept constant, “hopping” in and out of an intermediate can be observed. Such force clamp measurements report force-dependent transition rates directly, although typically in a limited force range. When the applied force is continuously varied, intermediates can be detected in the force-extension curves. Such force ramp measurements report the distribution P(F) of transition forces, which is a treasury of mechanistic information, but requires a theoretical foundation for this information to be extracted. When attempting to develop such a theoretical foundation, the first fact to face is that intermediates complicate the problem simply because they entail new types of transitions. Add to this a varying force, and the rates of all transitions become time-dependent. These complications, even in the presence of a single intermediate, give rise to interesting kinetic scenarios. One notable scenario8 occurs when the major energy barrier disappears under force and an initially minor barrier emerges to dominate the kinetics, all within a single measurement. Conformational transitions via a single intermediate, beside being ubiquitous in biology and nonbiological applications,1, 9 also provide insights into more complex a) A. Garai and Y. Zhang contributed equally to this work. b) Author to whom correspondence should be addressed. Electronic mail:
[email protected]. 0021-9606/2014/140(13)/135101/5/$30.00
dynamics. In this article, we show that the kinetics of a system with an intermediate, subject to a steadily increasing force [Fig. 1(a)], can be described analytically with a microscopic theory of a substantial generality. The theory culminates in an analytical expression for the transition force distribution P(F), an experimentally measurable quantity. The expression predicts the spectrum of behaviors that P(F) exhibits across a broad range of parameters, and reveals the emergence of distinct features [Fig. 1(b)]. These features provide a route to identifying the mechanism of a conformational transition among multiple candidate mechanisms. Through the established analytical relationship between the observable features and the microscopic parameters, the derived expression enables the extraction of these parameters (Fig. 2) from experimental data. By definition, an intermediate (I) is a metastable state separated from the native (N) and unfolded (U) states by significant energy barriers [Fig. 1(a)]. Consider an experiment in which a steadily increasing force is applied to a system with an intermediate, and the value of the force needed to transit into U is recorded. This transition force is determined by the parameters of the energy landscape, the force loading rate, and thermal noise. Due to the stochastic nature of the transition, the transition forces recorded in multiple repeats of this experiment are distributed according to a certain probability density P(F). Conformational changes through an intermediate under a force F(t) are governed by the master equation for the survival probabilities Si (t) in states i = {N, I, U}: ⎡˙ ⎤ ⎡ ⎤⎡ ⎤ SN (t) kI N (t) 0 SN (t) −kNI (t) ⎢ ˙ ⎥ ⎢ ⎥⎢ ⎥ ⎣ SI (t) ⎦=⎣ kNI (t) −kI N (t) − kI U (t) 0 ⎦⎣ SI (t) ⎦ S˙U (t)
0
kI U (t)
0
SU (t) (1)
While a single-intermediate system with constant rate coefficients kij is a textbook example of an analytically tractable kinetic description,9 the system in Eq. (1) is not: a non-trivial aspect of its kinetics is the time-varying perturbation due to the force F(t). The force introduces the time-dependence
140, 135101-1
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135101-2
Garai, Zhang, and Dudko
J. Chem. Phys. 140, 135101 (2014)
To convert Eq. (2) into a practical, closed-form expression, we must (i) determine the functional form of the individual force-dependent rates kij (F(t)), (ii) integrate these rates to find gij (F(t)), and (iii) perform the integration in Eq. (2). Step (i) is accomplished by applying Kramers formalism, which treats an elementary reaction as Brownian motion,10 to free energy profiles of different shapes captured by a scaling factor ν, and further noting that the pulling device is typically soft compared to the N and I states,11 κ 2kB T/x‡2 : ‡
FIG. 1. (a) Free energy profile with the native (N), intermediate (I), and unfolded/unbound (U) states, perturbed by a force F(t). (b) Transitions into U result in a distribution of transition forces, P(F), measurable in experiment. The analytical theory reveals how a bimodal (upper) or a unimodal (lower) P(F) emerge from the interplay between the population in I and the rate at which U receives population from I.
in the rate coefficients kij (t), thereby not only enriching the behavior of the probabilities Si (t)—but also making it challenging to solve Eq. (1) analytically. Nevertheless, we will now show that this system is amenable to a fairly general analytical treatment. The quantity of interest, the experimentally measurable distribution of forces at which the system transits into the final state (U), is defined as P (F ) ≡ kI U (F )SI (F )/F˙ , where F˙ = κV is the force loading rate, κ is the spring constant of the pulling device, and V the pulling speed. Because the external force lowers the barriers for forward transitions and raises the barriers for backward transitions [Fig. 1(a)], beyond very small forces, transitions I → N become rare: kIN ≈ 0. Expressing SI (F) and SN (F) as the respective formal solutions of the second and first equations in Eq. (1), we obtain the integral expression for the quantity of interest: kI U (F ) −gI U (F ) F gI U (f ) kNI (f ) −gNI (f ) e e P (F ) = e df, (2) F˙ f˙ 0 F where gij (F ) ≡ 0 kij (f )df/f˙ and the change of the variable of integration from time t to force F is straightforward due to a monotonic increase of F(t) prior to a transition. Note that a transition via an intermediate, unlike its simple single-barrier counterpart, involves a generally non-negligible drop in the force from f to f occurring in the course of the transition—upon escape from N to I—due to partial unfolding; Eq. (2) accounts for this drop.
‡
FIG. 2. Intrinsic parameters sought to be reconstructed: the heights Gij of the activation barriers, the locations diate, and kinetic rates kij0 .
‡ xij
of these barriers and of the interme-
⎡
Gij
⎣ ‡ 1 νF xij ν −1 kB T 1− 0 e kij (F ) = kij 1 − ‡ Gij
1−
‡ νF xij ‡ Gij
ν1 ⎤ ⎦
,
(3)
where kB is the Boltzmann constant and T the temperature. The integral of Eq. (3) with respect to force can be solved exactly, which readily takes us through step (ii): ‡
⎡
ij ⎣ G 1− kB T kB T kij0 e gij (F ) = ‡ F˙ xij
1−
‡ νF xij ‡ Gij
ν1 ⎤ ⎦
−1 .
(4)
To fulfill the final step (iii), we note a difference in behavior with respect to force of the two functions: a gradually decreasing e−gNI (F ) and a steeply increasing kI U (F )egI U (F ) (a special case where this behavior does not hold is analyzed separately). Then, the steepest descent method of evaluating the integral in Eq. (2) yields (see the supplementary material12 ) the sought-for analytical solution: 1 kI U (F )e−gI U (F ) [1 − e−gNI (F ) ] + kNI (F ) P (F ) = F˙ × e−gNI (F ) [1 − e−gI U (F ) (1 + gI U (F ))] 1 + gI−1 U (F ) (5)
with kij and gij given by Eqs. (3) and (4) and F = F − F accounting for the force drop F upon transition N → I, discussed above. The analytical expression in Eq. (5) for the experimentally measurable distribution of transition forces in a system with an intermediate is the key result of this paper. The predictive power of Eq. (5) is explored in Fig. 3 through a systematic overview of representative free energy landscape topologies possessing an intermediate. Equation (5) is applicable to topologies I, II, III, and V; a discussion of topology IV will follow later. The transition state (‡) is defined as the major barrier in the absence of force. The resulting kinetic scenarios under a force-ramp can be understood in terms of two key properties of the intermediate I. First, the location of I relative to ‡: before (“early intermediate”) or after (“late intermediate”). Second, the pliability of I relative to that ‡ ‡ of the native state N: a soft I implies xI U > xNI , a stiff I implies the inverse. For an early and stiff intermediate (case II in Fig. 3) and for a late and soft intermediate (case III), the kinetics at all forces are dominated by the transition state. However, a switch in dominance from the transition state to the initially minor barrier occurs in two topologies: an early and soft intermediate (case I) and a late and stiff intermediate (case IV). When two comparable barriers are separated by an intermediate of a pliability similar to that of state N (case V), both barriers equally dominate the kinetics at all forces.
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135101-3
Garai, Zhang, and Dudko
J. Chem. Phys. 140, 135101 (2014)
FIG. 3. A systematic overview of force-induced transitions through an intermediate. (a) Representative topologies (I-V) of the free energy profile with an intermediate at three values of force. (b) Representative force-extension curves at low, medium, and high force loading rate κV . Arrows indicate the measured rupture forces. (c) Rupture force distributions P(F) for the transition into the final state (U) at three values of κV . Histograms are simulated data, lines are a global fit to the analytical theory (see the supplementary material12 for details on cases III and IV). (d) Most probable transition force Fpeak vs. loading rate (the force spectrum) for two types of transitions: I → U (triangles, these are the peaks of the histograms in (c)) and N → I (circles). Black/gray lines are the peak peak predictions of Eq. (8), F peak = FI U (κV ) and F peak = FNI (κV ) − F , respectively, if the outer/inner barrier were to dominate at all κV . Note that the triangles follow the black/gray line whenever the outer/inner barrier is dominating. The triangles switch between the black and gray lines in cases I and IV where the switch in the dominant barriers occurs. (e) Individual force-dependent rates, Eq. (3), for transitions N → I, I → U, and I → N, and the effective rate NI (F )kI U (F ) keff (F ) = kNI (Fk)+k . Beyond very low forces, the effective rate follows the rate limited by the dominant barrier. I N (F )+kI U (F )
Equation (5) further reveals that the presence of an intermediate may make the transition force distribution undergo a transformation from unimodal to bimodal and back to unimodal when increasing the loading rate (Fig. 3(c), case I). The possibility of a bimodal force distribution due to an intermediate has been pointed out by Strunz et al.13 who simulated the master equation, Eq. (1), numerically. We now show how the bimodality in the force distribution emerges, as a transient scenario, from the general analytical solution [Eq. (5)] to the master equation. What is the origin of the bimodality in the transition force distribution? Equation (5) shows that the condition for bimodality is an early and soft intermediate (case I in Fig. 3). At short times, and hence low forces, population accumulates in I, blocked from entering U by a high transition state (Fig. 4). However, because the transition state is soft and thus lowered rapidly, I soon begins to release this accumulated population (hence the increase in the measured P(F)), while the rest of the population remains trapped in the stiff N. Once
the population in I is depleted, a drop in P(F) occurs, completing the first peak. This short-time behavior is captured by the term kI U (F )e−gI U (F ) in Eq. (5). At long times, and hence high forces, the stiff inner barrier is lowered sufficiently for N to release its trapped population, leading to a new rise in the measured P(F). Once the remaining population is released from N, P(F) drops again, completing the second peak. This long-time behavior is captured by the term kNI (F )e−gNI (F ) in Eq. (5). Remarkably, Eq. (5) reveals that the bimodality caused by the presence of an intermediate is always described by a particular configuration of the two peaks, namely a narrow peak at lower forces followed by a broader peak at higher forces (Fig. 3(c), case I). This distinct, and persistent, configuration of the two peaks is a consequence of the soft nature of I relative to that of N, a condition necessary for the bimodality to emerge. Indeed, a transition from a softer state (here, I) will lead to a more narrow distribution (here, first peak) of transition forces, because a soft state rapidly releases its population,
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135101-4
Garai, Zhang, and Dudko
J. Chem. Phys. 140, 135101 (2014) peak where F = F + F − F˙ τ with τ ≡ kI−1 U (FNI (κV ) − F ) accounts for a drop in the force from F in state N to (F − F˙ τ ) upon escape into I, and a subsequent increase to the force F (the measured value) upon escape into U. peak The value of FNI (κV ) is given by Eq. (8) with α = 1 and ij = NI. The most probable force, Fpeak , in the force distribution P(F) for the transition I → U is described by (see the supplementary material12 ) ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ peak ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎩
Fij
dip med
⎪ ⎪ ⎪ ⎪ ⎭
‡
(κV ) =
Gij ‡
νxij
ν ‡ ακV xij ln 1− 1− . ‡ kB T kij0 Gij
kB T
(8)
FIG. 4. The origin of bimodality in the transition force distribution of a system with an intermediate. The distribution, P(F), is equivalent to the flux into the final state (U), given by the rate (set by the barrier) times the population. The arrow widths reflect the fluxes at different instants.
giving less time for thermal fluctuations to broaden the resulting force distribution. The described distinct features of the bimodality in the force distribution can be used as a basis for identifying or eliminating the intermediate as the mechanism of a conformational transition among several candidate mechanisms. For topology IV, the expressions for P(F), derived in the supplementary material12 , are as follows. At high loading rates, where the most probable forces for transitions N → I and I → U begin to diverge (Fig. 3(d), case IV), P (F )
1 gI U (F med ) e [1 − e−gNI (F ) ]kI U (F )e−gI U (F ) , F˙
(6)
med (κV ) − F with where F = F + F and F med = FNI med FNI (κV ) being the median force for transition N → I [given by Eq. (8) with α = ln 2 and ij = NI]. At low loading rates (but high enough for transitions I → N to be negligible), where the most probable forces for transitions N → I and I → U are still close,
P (F )
1 kNI (F )e−gNI (F ) , ˙ F
(7)
When the force distribution for I → U is bimodal, the locations of the first and second peaks can be found from peak peak Eq. (8) as F peak1 = FI U (κV ) and F peak2 = FNI (κV ) − F with α = 1. We further find (see supporting material12 ) that Eq. (8) also captures the location of the dip in the dip bimodal distribution as F dip = FI U (κV ) with α = 8. When the force distribution for I → U is unimodal, the location of its peak can be found from Eq. (8) as F peak peak = FNI (κV ) − F if the first barrier is dominating and as peak F peak = FI U (κV ) if the second barrier is dominating, with α = 1. Thus, Eq. (8) unifies, via α, a set of relationships between observable features (the peaks and the dip in the force distribution) and the parameters of the system (Fig. 2) and, as such, provides additional means for extracting these parameters. The performance of the developed theory as a fitting tool is reported in Fig. 3 and Table I with data from Brownian dynamics simulations of systems with an early and soft intermediate (case I in Fig. 3) at parameters representative of the optical tweezers and AFM setups. Table SI reports the fitting results for all other topologies (cases II-V in Fig. 3) (see the supplementary material12 ). In principle, a fit of the force histograms for the final transition (I → U) to Eq. (5) in a broad range of loading rates recovers the heights and locations of the two barriers and the corresponding transition rates. In practice, however, an insufficient quality of the histograms may affect the robustness of such a six-parameter fit, and the following two-step procedure may prove more efficient. First, force histograms for the first transition, N → I, at different values of κV are fitted globally (i.e., with a unique set of three fitting parameters for all histograms) to the singlebarrier expression pNI (F ) = F1˙ kNI (F )e−gNI (F )11 with kNI (F) and gNI (F) in Eqs. (3) and (4) at a fixed ν (0.5 in Fig. 3). This
TABLE I. Kinetic and energetic parameters of a system with an intermediate (Fig. 2), extracted from the fit of the simulated data to the derived analytical theory. The values of the stiffness κ (in pN/nm) represent optical tweezers and an AFM. ‡
True, κ= 0.1 Fit True, κ= 5 Fit
‡
‡
‡
‡
‡
xNI [nm]
GNI [kB T]
0 ln(kNI [s−1 ])
xI N [nm]
GI N [kB T]
ln(kI0N [s−1 ])
xI U [nm]
GI U [kB T]
ln(kI0U [s−1 ])
1.00 1.08 ± 0.02 0.30 0.31 ± 0.01
16.0 15.8 ± 0.9 18.0 18.7 ± 2.6
− 4.5 − 4.6 ± 0.1 − 4.4 − 3.9 ± 0.1
1.80 1.72 ± 0.02 0.60 0.59 ± 0.01
8.0 9.7 ± 0.8 8.0 11.6 ± 2.8
2.2 1.5 ± 0.1 4.2 3.2 ± 0.1
4.5 4.5 ± 0.6 1.5 1.8 ± 0.3
22.0 26.4 ± 3.5 25.0 26.6 ± 0.9
− 13.2 − 13.5 ± 1.1 − 15.2 − 16.3 ± 1.6
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135101-5
Garai, Zhang, and Dudko ‡
J. Chem. Phys. 140, 135101 (2014) ‡
fit yields the height GNI and location xNI of the first barrier 0 . Next, force histograms and the associated intrinsic rate kNI for the second transition, I → U, are globally fitted to Eq. (5) ‡ ‡ 0 fixed at the values extracted from with GNI , xNI , and kNI the fit of the first transition. The average drop in the force, F, is read off the force-extension curves (see the supplementary ‡ ‡ material12 ). This fit yields the height GI U and location xI U 0 of the second barrier relative to I and the intrinsic rate kI U . To determine whether Eq. (5) is applicable to the system under study, we locate the peaks in the force histograms for N → I and I → U (circles and triangles, respectively, in Fig. 3(d)) to determine the most probable forces for these transitions, and plot these forces together as a function of the loading rate κV . If these forces are close at low κV and diverge with increasing κV , which matches the pattern of the force spectrum for case IV in Fig. 3(d), then Eq. (6) can be used to fit the force histograms for I → U at high κV and Eq. (7) at low κV . In all other cases, Eq. (5) can be used to fit the force histograms for I → U in the entire κV range. The remaining parameters—the location of the intermediate I, the intrinsic rate for transition I → N and the activation barrier separating state I from state N (Fig. 2)—can be estimated as follows (see the supplementary material12 ): ‡
‡
‡
xI N ≈ F /κeff − xNI , kI0N ≈ kI N (F1/2 )eF1/2 xI N /kB T , (9) 0 ‡ ‡ − ln kI0N , GI N ≈ GNI + kB T ln kNI where the mid-point force F1/2 and kIN (F1/2 ) can be determined from a force-clamp measurement and κ eff is the effective spring constant of the pulling device and the linker (if present) tethering the molecule to the device. The analytical framework developed above should be used with caution at very high and very low forces. At very high forces, a limited time resolution may prevent the detection of the intermediate in the force extension curves; if in-
creasing the time resolution is not possible, these data sets should not be included in the fit. At very low forces, the assumptions of the theory may no longer be justified if backward transitions I → N become non-negligible or if the drop in the force (F) upon transition N → I varies significantly from its average. The developed theory unifies a broad range of macromolecular systems and experimental techniques. The established general principles provide a quantitative framework for mechanistic understanding of multistep conformational transitions and may point to universal strategies for influencing specific steps in the related processes. This research was supported by NSF CAREER Grant No. MCB-0845099 and NSF CTBP Grant No. PHY-0822283. 1 G.
M. Whitesides, J. P. Mathias, and C. T. Seto, Science 254, 1312 (1991). 2 B. Alberts et al., Molecular Biology of the Cell (Garland Science, New York, 2008). 3 M. Vendruscolo and C. M. Dobson, Nat. Chem. Biol. 9, 216 (2013). 4 P. E. Marszalek, H. Lu, H. Li, M. Carrion-Vazquez, A. F. Oberhauser, K. Schulten, and Julio M. Fernandez, Nature (London) 402, 100 (1999). 5 C. Cecconi, E. A. Shank, C. Bustamante, and S. Marqusee, Science 309, 2057 (2005). 6 M. T. Woodside, P. C. Anthony, W. M. Behnke-Parks, K. Larizadeh, D. Herschlag, and S. M. Block, Science 314, 1001 (2006). 7 J. Stigler, F. Ziegler, A. Gieseke, J. C. M. Gebhardt, and M. Rief, Science 334, 512 (2011). 8 R. Merkel, P. Nassoy, A. Leung, K. Ritchie, and E. Evans, Nature (London) 397, 50 (1999). 9 A. R. Fersht, Structure and Mechanism in Protein Science (W. H. Freeman, New York, 1999). 10 H. Kramers, Physica 7, 284 (1940). 11 O. K. Dudko, G. Hummer, and A. Szabo, Phys. Rev. Lett. 96, 108101 (2006). 12 See supplementary material at http://dx.doi.org/10.1063/1.4869869 for details on the derivations and simulations. 13 T. Strunz, K. Oroszlan, I. Schumakovitch, H.-J. Güntherodt, and M. Hegner, Biophys. J. 79, 1206 (2000).
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