Bell 's expression and the generalized garg form for forced dissociation of a biomolecular complex

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(1)PRL 98, 088304 (2007). PHYSICAL REVIEW LETTERS. week ending 23 FEBRUARY 2007. Bell’s Expression and the Generalized Garg Form for Forced Dissociation of a Biomolecular Complex Han-Jou Lin,1 Hsuan-Yi Chen,2,* Yu-Jane Sheng,1,* and Heng-Kwong Tsao3,* 1. Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106, Republic of China Department of Physics, Graduate Institute of Biophysics, Center for Complex Systems, National Central University, Jhongli, Taiwan 320, Republic of China 3 Department of Chemical and Materials Engineering, National Central University, Jhongli, Taiwan 320, Republic of China (Received 27 June 2006; published 23 February 2007) 2. The dissociation of a biomolecular complex under the action of constant force, constant loading rate, and periodic force is studied theoretically. We show that the celebrated Bell expression provides a good approximation for the bond dissociation rate when F=Fc 1, where Fc is the maxima slope of the binding potential along the reaction coordinate. When 1 F=Fc 1 the dissociation rate is better described by a generalized Garg form in which the potential derivative is expanded near Fc . We also show that a constant-force experiment is suitable for extracting the activation energy of the bond, a constant loading experiment is suitable to extract Fc , while time-periodic force can be applied to extract both bond dissociation rates at zero force and Fc . DOI: 10.1103/PhysRevLett.98.088304. PACS numbers: 82.37.Np, 87.15.v. The dynamic strength of noncovalent biomolecular bonds has been experimentally explored by singlemolecule pulling experiments [1,2]. In these constant loading rate experiments, the strength of the bond is characterized by the relation between the rupture force Fu and loading rate F_ t , i.e., the dynamic force spectroscopy. The main issue is how single-molecule pulling experiments should be analyzed to extract useful information associated with the kinetics and energy landscape along the dissociation pathway [3,4]. The theoretical problem of finding the rupture force of a bond under constant loading rate corresponds to evaluating the escape time of a particle trapped in a one-dimensional potential well in the presence of both a time-ramped external force and thermal fluctuations, an interesting extension of the celebrated Kramers rate theory [5,6]. The primary role of external pulling in the dissociation kinetics is lowering the energy barrier Ea . Following the seminal work of Bell [7], it is often assumed that the metastable potential well is deep and the force-dependent energy barrier is Ea F E0a Fa, where E0a represents the intrinsic activation energy and a the distance between the potential well and the barrier along the reaction coordinate. Up to now, Bell’s expression is regarded as a phenomenological theory which is convenient to extract the intrinsic dissociation rate constant k0 and a from pulling experiments [3,4]. However, another work by Garg [8] assumed that bond rupture occurs when the metastable well becomes very shallow; in this case a cubic function is appropriate to approximate the potential surface in the vicinity of the critical force (Fc ) and Ea F E0a 1 F=Fc 3=2 . Bell’s expression yields Fu lnF_ t , while the Garg form results in Fu lnF_ t 2=3 . Recently a unified form with a fitting parameter [4] is proposed to extract k0 , a, and E0a . In this approach 1 the escape rate under force F is kF; F; 0031-9007=07=98(8)=088304(4). expEa F; , where the activation energy is 1 k0 1 Ea F; E0a 1 Fa=E0a 1= , and F; 0 1=1 E0a Fa=Ea e . Bell’s expression is recovered for 1 or E0a ! 1, while the Garg form is obtained for 2=3. It was suggested that this form is more general than Bell’s expression. On the other hand, it has also been pointed out [3,9] that for a single-well potential, Bell’s expression and the Garg form are actually valid in different regimes. They correspond to two asymptotic limits: (i) the slow pulling regime (Fu =Fc 1) and (ii) the fast pulling regime (1 Fu =Fc 1), respectively. Reference [3] further confirmed their claim by fitting atomic force microscope and biomembrane force probe measurement for unbinding biotinstreptavidin complex [2] to both limits. To settle this issue and to provide a theoretical foundation for interpreting experimental data, in this Letter we derive Bell’s expression and the generalized Garg form from Kramers rate theory and show that both of them are asymptotic expressions. The former is valid for weak pulling (F=Fc 1), and the latter is valid for strong pulling (1 F=Fc 1). We also discuss the bond lifetime in constant force, constant loading rate, and periodic force experiments. Our study shows that useful information about the bond can be extracted by performing different types of pulling experiment. Barrier crossing under external force.—Consider an overdamped Brownian particle escaping from a metastable well of Ux under the action of force F, where x is the reaction coordinate. The dynamics of this particle is described by the Langevin equation M dx dt d Ux Fx t. The friction coefficient M is dx related to the diffusivity D by M kB T=D. M comes from the dissipation of kinetic energy into internal degrees of freedom. Therefore, M in the trapped state may be significantly greater than that in the escaped state; in this case rebinding events can be neglected.. 088304-1. © 2007 The American Physical Society.

(2) PHYSICAL REVIEW LETTERS. PRL 98, 088304 (2007). Thermal force has zero mean and htt0 i 2MkB Tt t0 . The free energy landscape associated with Ux is characterized by three parameters: the interaction range a x0 x0 which denotes the distance from the metastable well x0 to the barrier x0 , the intrinsic energy barrier E0a Ux0 Ux0 , and the maximum slope of the trapping potential, i.e., the critical force, Fc U0 xc . Typically E0a O10kB T, a O1 nm, and Fc O10 pN. When F 0, the biomolecular complex dissociation occurs due to thermal fluctuations. For Ea kB T, the escape rate is given by the Kramers theory [5,6]: k 1 expEa ;. with. . 2 : ! !. (1). Here !

(3) U00 x

(4) =M 1=2 . This thermally activated escape denotes the kinetic limit. On the contrary, when F > Fc the binding force can be overcome without any help of thermal fluctuations. The dissociation is dominated by mechanical pulling; this is the mechanical limit. When 0 < F < Fc , the external force alters the effective trapping potential to Vx Ux Fx. Thus, the well and saddle positions are determined by U0 x

(5) F. The barrier height becomes Ea F Vx F Vx F , becomes F, and the rate constant at F 0 is k0 E0a . Because the energy barrier is lowered, the 1 0 e bond lifetime is reduced with increasing F. Simple analytical expressions for the bond lifetime under constant force can be obtained in the asymptotic limits F=Fc 1 and 1 F=Fc 1. When F=Fc 1, x

(6) is close to x0

(7) , and the trapping potential near x0

(8) can be approximated by Ux Ux0

(9) 12 U00 x0

(10) x x0

(11) 2 since U0 x0

(12) 0. Therefore the new positions of well and saddle point are x

(13) F x0

(14) U00Fx0 . The force

(15). dependent barrier height is then given by Ea F Vx Vx F2 1 1 0 Ea Fa. 2 U00 x0 U00 x0 . (2). and U00 x

(16) U00 x0

(17) U3 x0

(18) U00Fx0 . The leading con

(19). tribution to Ea F and U00 x

(20) leads to an escape rate kF k0 eFa ; this is the celebrated Bell’s expression [7]. The basic assumption in the above derivation is that the well and barrier are only slightly shifted away from the original positions. Consequently, Bell’s expression is valid when the pulling force is weak. When 1 F=Fc 1, x

(21) is in the vicinity of xc ; therefore, the dissociation rate of the bond can be found by expanding Ux near xc . Since U00 xc 0 and U0 xc Fc , one has U0 x Fc U3 xc x xc 2 =2 at x close to xc . If U3 xc 0, then one needs to expand to higher order and U0 x Fc Un 1 xc x xc n =n!. Here n > 2 is even because U0 x has maximum value at xc . This corresponds to the following approximation for the poten-. week ending 23 FEBRUARY 2007. tial: U F2c b y yn 1 =n 1 Uxc , where y x xc =b=2 and b 2n!Fc =Un 1 xc 1=n is the characteristic length of U in this regime. The well and barrier for V U Fx are therefore located at x

(22) F

(23) b2 1 F=Fc 1=n xc . The barrier height Ea F and the intrinsic time scale F in this regime are expressed by F n 1=n Ea F Vx Vx Ea 1 Fc (3) 1 2M 1 F n1=n p and ; 1 F Fc jU00 x U00 x j n where Ea n 1 bFc and 1 n 1Ea =b2 =D . 1 Although inserting Eq. (3) into kF F expEa F with n 1=n yields an expression similar to the general form in [4], our theory actually is very different from Ref. [4]. The general form in [4] is not derived from the true potential Ux; it is a proposed form with fitting parameters E0a , a, and k0 . Like any reasonable approximation for Ux, this expression reduces to Bell’s expression in small F limit. Reference [4] suggests that by fitting the experimental data to their general form a better fit for E0a , a, and k0 , compared to Bell’s phenomenological theory, can be obtained, and it does not matter whether the experiment is carried out under F Fc or 1 F=Fc 1. However, we show that Bell’s expression represents a small perturbation to the thermally activated escape caused by weak pulling. As a result, the intrinsic properties such as k0 and a are preserved in Bell’s expression but the information about the critical force Fc is beyond Bell’s expression. On the other hand, when 1 F=Fc 1, the apparent free energy landscape is significantly altered, and thereby k0 and a cannot be obtained from the escape rate. In this case the apparent energy barrier is proportional to 1 F 3=2 . This indicates that one can extract Fc from strong Fc pulling experiments. Notice that Fc is an important feature of the potential that has been mostly ignored so far. In vivo, the rupture of soft bonds can be thermally assisted or mechanically induced; in the latter case Fc is the relevant quantity, not k0 or a. To examine our analysis, we consider a model potential E0 x ; 0 x a: (4) Ux a 1 cos a 2. The escape rate k for this potential under constant force can be obtained analytically. Straightforward algebra 0 F leads to x a sin1 Fc , x a x , and Fc E2a a. The energy barrier and the intrinsic time scale are given q 1 by Ea F E0a 1 FFc 2 FFc cos1 FFc and F F 2 1=2 , respectively. In the limit F=Fc 1, 1 0 1 Fc 1 0 one has Ea Ea 1 2 FFc 12 FFc 2 E0a Fa and F 1 F 2 1 , consistent with Bell’s expression. 1 . 1 0 0 2 Fc p In the limit 1 F=Fc 1, one has Ea 2 3 2 E0a 1 FFc 3=2. 088304-2.

(24) week ending PHYSICAL REVIEW LETTERS 23 FEBRUARY 2007 PRL 98, 088304 (2007) p 1 F 1=2 be done. When hTi p , in the small amplitude regime and F 21 , consistent with the general0 1 Fc F0 =Fc 1 Bell’s expression is valid, and by substituting ized Garg form with n 2. This confirms that Bell’s and Ft into Bell’s expression one finds Garg’s expressions are valid in different regimes. Constant loading rate and periodic forcing experihTi k0 I0 F0 a 1 ; (8) ments.—Now we consider experiments with constant loading rate, Ft F_ t t. Although Ea and vary with time, the for F0 a 4. In the large amplitude regime 1 s F ktP t if the survival probability Ps t satisfies dP 0 =Fc 1, the lifetime is obtained by applying the gens dt eralized Garg form Eq. (3): Brownian particle is adjusted to the apparent potential, s Ux Ftx, instantaneously. Therefore Ps t Rt 0 0 1 n 1 F0 1=n F0 exp 0 kt dt , and the mean bond lifetime hTi E 2 hTi ; (9) 1 a R1 dPs kF0 n Fc Fc t dt is given by 0 dt Zt where kF0 F1 0 expEa F0 denotes the rate conZ1 0 0 hTi t kt exp kt dt dt; (5) stant associated with constant pulling F0 . Therefore in both 0 0 limits hTi increases with F0 but is independent of p . The and the rupture force Fu F_ t hTi. Analytical expressions lifetime is independent of the frequency of the force for the for hTi can be obtained for the slow pulling F_ t hTi Fc following reason. In each period, the escape probability is and fast pulling 1 F_ t hTi=Fc 1 regimes. In the slow pulling regime, the bond lifetime is obtained by substitut10 ing Bell’s expression kFt k0 eF_ t at into Eq. (5). The n=1 n=2 resulting bond lifetime is n=4 Bell's expression 1 k0 k0 10 hTi exp E1 : (6) aF_ t aF_ t aF_ t 6 R1 eu du 10 Here E1 x x u is the exponential integral. When 4 _Ft a=k0 kB T, the change of energy barrier height due to external force during time interval 1=k0 is greater than 10 2 thermal energy. Thus bond lifetime is affected by the pulling, and the above expression can be shown to be reduced 0 1 2 _ 1 Fa / kT to [3] F_ t hTi a lnak0Ft ; here 0:5772 . . . is the 10 0 4 8 12 16 20 Euler constant. Thus the rupture force grows logarithmiFa / kT 1.0 cally with F_ t in the slow pulling regime. In the fast pulling 3 regime, bond dissociation occurs when the ramped pulling approaches Fc . Inserting the approximate potential Eq. (3) 2 0.8 into Eq. (1), then performing asymptotic integration in 1 Eq. (5) for Ea F 1, yields the generalized Garg 0.6 form for bond lifetime [8]: 0 10 10 F / (D/ a ) Fc lnbF_ t Ea n2=n 1 n=n 1 0.4 hTi 1 : (7) Ea F_ t 4. 2. ⟨k ⟩ / k0'. ⟨k ⟩ /k0. 3. 1. F ua / kT. 0. Fu / Fc. -3. For the most general case n 2, rupture force scales as lnF_ t 3=2 . From the above analysis, the much debated question of whether Bell’s expression or the generalized Garg form is appropriate to describe constant loading rate experiments [4] becomes clear. One should first chose either Bell’s expression or the generalized Garg form to estimate Fc from the experimental data, then check whether the experiment is conducted in the weak pulling or strong pulling limit by comparing the rupture force to Fc . It is not true that one expression is better than another in all experiments, but it is possible that many existing experimental data are actually taken from experiments conducted in the strong pulling limit [4]. Experiments on the dissociation of a single bond under time-periodic force Ft F0 cos2t=p can also. -2. 3. t. n=1 n=2 n=4 Eq. (6). 0.2. (b) 0.0 -4 10. 10. -3. 10. -2. 10. -1. 10. 0. 10. 1. 10. 2. 3. Ft / (D/ a ). FIG. 1 (color online). Comparisons between asymptotic expressions and Langevin dynamics simulations for U0 x f1 jx xc j=b=2 n gn 1E0a =nb with n 1, 2, and 4. The activation energy for different potentials is the same, E0a 10. (a) Constant-force escape for hki hTi1 and (b) constant loading rate escape for Fu =Fc . The dashed lines depict the expressions based on the generalized Garg form [Eqs. (3) and (7)]. The insets show (a) the weak force regime and (b) the slow pulling regime, where Bell’s expression [Eqs. (2) and (6)] is valid. The intrinsic rate constant k00 is determined by simulations.. 088304-3.

(25) PHYSICAL REVIEW LETTERS. PRL 98, 088304 (2007) 5. 10. 4. 10. 3. 10. 2. 10. 1. Eq. (5) Eq. (8) Eq. (9). 10. ⟨T⟩ / ⟨T⟩ 0. ⟨k ⟩ / k0. 10. 10. 0. =-. /2. -2. =0 -4. 10 -3 10. 10. week ending 23 FEBRUARY 2007. 10. 0. 10. 3. 10. 6. p. 0. 0. 4. 8. 12. 16. 20. F0a / kT. FIG. 2 (color online). Bond lifetime under time-periodic force Ft F0 cos2t=p in the p -independent regime. hTi0 denotes the intrinsic bond lifetime for Ux 61 cosx=a . The lines represent numerical solution of Kramers rate theory [Eq. (5)] and asymptotic expressions [Eqs. (8) and (9). Since we have adopted a periodic Ux for simulations, hTi0 k1 0 evaluated by Eq. (1) is about twice greater than that obtained from simulations.. largest in the vicinity of F0 , and this escape window is p . Since the occurring frequency of this window is proportional to 1 p , the total escape window over many periods is independent of p as long as hTi p . Discussion.—To illustrate the ideas presented in our previous discussion, consider potentials with U0 x f1 jx xc j=b=2 n gn 1E0a =nb for n 1, 2, and 4. The intrinsic barrier height is E0a , the critical force for given n is Fc n 1E0a =nb, and different n corresponds to different potential shape. Langevin dynamics simulations are performed for E0a 10 to get k for constant force [Fig. 1(a)] and Fu for constant loading rate [Fig. 1(b)] escape. In Fig. 1, simulation results are compared to results from Bell’s and Garg’s expressions. The insets demonstrate the validity of Bell’s expression in weak (constant force) and slow (constant loading rate) pulling regimes; both Figs. 1(a) and 1(b) show that the generalized Garg forms agree with the simulation results over a wide range of external forces and loading rates. From Fig. 1(a) we also find that it is difficult to distinguish constant-force simulation results for different choices of n with the same E0a 10, except when the external force is very close to or exceeds the Fc of some chosen n. This suggests that bond lifetime in most constant-force experiment is dominated by the intrinsic barrier E0a , not by the details of the potential surface (in this case, n or Fc ). On the other hand, distinct Fu results are obtained in Fig. 1(b) for different choices of n. Under the same loading rate, the rupture force increases with n. Thus the constant-force pulling experiment is more useful in extracting E0a than the constant loading rate one, while it has been pointed out [3,9] that the latter is suitable to probe the details of the free energy landscape close to Fc .. Langevin dynamics simulations are also performed for periodic forcing for the potential given by Eq. (4). As shown in the inset of Fig. 2, at very small p the adiabatic approximation is not valid and the lifetime approaches a constant value independent of F0 , which is not of practical experimental interest. At very large p ( hTi), the bond lifetime approaches another constant for 0 because the bond is effectively under constant force F0 cos when it breaks. For =2, the bond lifetime at very large p becomes the constant loading rate lifetime with F_ t 2F0 =p . In the intermediate p where the adiabatic approximation is valid and still p hTi, there is indeed a region where hTi is independent of p and Eq. (8) or (9) can apply, depending on the magnitude of F0 . Figure 2 shows hki hTi1 as a function of F0 for p -independent region simulations. Equation (9), based on a general Garg form, describes hki with high precision at large F0 , while Eq. (8), based on Bell’s expression, describes hki well at small F0 (data not shown). This consequence indicates that periodic forcing experiments allow one to extract k0 and a at small F0 , and Fc and Ea at large F0 . In summary, we have derived both Bell’s expression and the generalized Garg form for forced escaped problem; they are approximations to describe single-molecule pulling experiments in different regimes. Our study suggests that the constant-force experiment is useful for extracting the intrinsic barrier height of the potential, the constant loading rate experiment is useful to study the shape of free energy landscape close to the inflection point, and the periodic forcing experiment can be used to find the intrinsic dissociation rate and the value of critical force. Therefore new ways to probe the inner lives of biomolecular complexes are revealed, and they should be easily checked in future experiments. This work is supported by the National Science Council of Taiwan.. *Corresponding author. [1] E. L. Florin, V. T. Moy, and H. E. Gaub, Science 264, 415 (1994). [2] R. Merkel, P. Nassoy, A. Leung, K. Ritchie, and E. Evans, Nature (London) 397, 50 (1999); Y.-S. Lo, Y.-J. Zhu, and T. P. Beebe, Jr., Langmuir 17, 3741 (2001). [3] Y.-J. Sheng, S. Jiang, and H.-K. Tsao, J. Chem. Phys. 123, 091102 (2005). [4] O. Dudko, G. Hummer, and A. Szabo, Phys. Rev. Lett. 96, 108101 (2006). [5] H. A. Kramers, Physica (Amsterdam) 7, 284 (1940). [6] P. Ha¨nggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990). [7] G. I. Bell, Science 200, 618 (1978). [8] A. Garg, Phys. Rev. B 51, 15 592 (1995). [9] H.-Y. Chen and Y.-P. Chu, Phys. Rev. E 71, 010901(R) (2005).. 088304-4.

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