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pH dependent protein stability: A quantitative approach based on Kramer's barrier escape
Chemical Physics Letters 618 (2015) 94–98
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
pH dependent protein stability: A quantitative approach based on Kramer’s barrier escape Debarati Chatterjee Center for Theoretical Biological Physics, University of California, La Jolla, San Diego, CA 92093, USA
a r t i c l e
i n f o
Article history: Received 27 July 2014 In final form 3 November 2014 Available online 8 November 2014
a b s t r a c t The folding–unfolding mechanism of a protein in a living cell is the result of its molecular response to various perturbations due to changes in the chemical environment such as pH, ion concentration, ligand binding and many other complex chemical reactions. In this letter, a theoretical scheme based on single molecule force spectroscopy experiments and related theories, has been followed to capture the effect of the change of pH for a slow protonation-coupled folding–unfolding process, and the free energy landscape has been quantified in terms of protonation numbers, barrier height and pH, following Kramer’s barrier crossing formalism. © 2014 Elsevier B.V. All rights reserved.
1. Introduction In a living cell, the mechanical response and folding–unfolding mechanism of a biopolymer is often an outcome of a series of chemical reactions. Thus a change in the folding kinetics due to the influence of any other chemical agent or ions is an important aspect of the folding mechanism. Many of the protein folding–unfolding kinetics are found to be triggered by Ca2+ , H+ or other ion concentrations [1–4]. These ions are found to play an important role in maintaining a well regulated physiologically viable timescale of the cell machinery by modulating the rate of the folding–unfolding kinetics of the involved proteins. Single molecule experiments based on the FRET (flouroscence resonance energy transfer) have also monitored the folding–unfolding process when they are triggered by chemical denaturants [5]. Thus, how the surrounding ions can change the stability of the protein is a thought provoking area of study in the field of biophysical research. Recent developments in various single molecule force spectroscopy techniques have enabled us to acquire a quantitative molecular view of the protein folding kinetics [6–9]. In this method, a biopolymer is probed by an externally applied force and the mechanical response of the biopolymer is monitored. The intrinsic free energy landscape of the biopolymer can be extracted in terms of its end-to-end distance from the related theoretical framework [10,11]. However, the effect of chemical activation on the folding unfolding process in still not satisfactorily explained in detail. Moreover, in a complex protein folding scenario, often the force spectroscopic data shows some
E-mail address: [email protected] http://dx.doi.org/10.1016/j.cplett.2014.11.002 0009-2614/© 2014 Elsevier B.V. All rights reserved.
unusual behavior. This is due to the involvement of other slower variables rather than end-to-end distance among the available vast number of degrees of freedom, originating from the biomolecule and its surroundings. In that case, only the incorporation of the proper reaction coordinate along the rate limiting variable, seems to provide an adequate description of the overall process [12,13]. In this letter, we highlight a folding scenario where the folding–unfolding event of the protein is pH dependent and we assume here that the process can be described by a slow protonation-coupled folding pathway. In the case of this slow protonation assisted protein folding, we propose here that the effect of the rate enhancement of the unfolding process by pH may be a direct effect of activated conformational change of the protein machinery due to an increase in [H+ ] concentration. Moreover, this pH-induced rapid conformational change of protein may resemble the scenario where the unfolding rate of a protein increases due to an external pulling force in a single molecule pulling experiment. The Dudko–Hummer–Szabo theory [10] provides an analytical method and an exact result based on the concept of Kramers’ escape [14] over a single barrier to quantify the activation barrier and the transition distance, from the unfolding rate vs. force plot obtained from experiments. The theory considers the unfolding process under an applied external force as a single barrier crossing process with respect to the end-to-end distance of the protein as the reaction coordinate. We believe, that a similar approach should be applicable here to extract the various quantities of interest from the pH dependent folding–unfolding experiment. Hence, on the basis of the above concept, considering the protonation number as the reaction coordinate, we aim to deduce a general theoretical background for the process of conformational change and provide a method to
D. Chatterjee / Chemical Physics Letters 618 (2015) 94–98
95
quantify the system in terms of its intrinsic parameters like rate of the process, barrier height and barrier distance, at the usual physiological condition, pH0 = 7. Thus, finding and quantifying the hidden free energy landscape of the unfolding event by simply probing the protein with pH via single molecule study is the main aim of this letter. 2. Theoretical background Let us consider a situation when, in the presence of an external pH, the folding rate of a protein increases compared to the folding rate of the same protein at a normal cellular pH value of pH0 = 7. In order to visualize the effect of the externally applied pH, we assume here that protonations activate the protein machinery on the folding pathway and we describe this overall transition of proton-assisted folding of the protein as a thermally activated single barrier escape problem. When the pH is set to be lower than pH0 = 7, it helps to lower the barrier of the folding pathway and the rate becomes faster. Thus, here, H+ ions trigger the conformational change in the protein. We formulate the free energy landscape U, where we choose the reaction coordinate as the number of added extra protons nH+ (t) (with respect to the average number of protons associated to the protein at pH0 = 7) getting bound to the proteins at any instant to trigger the conformational change at a certain pH under a constant temperature. This facilitates the transition from the unfolded to the final folded state. nH+ (t) at any instant, is a thermally fluctuating quantity, and is a random variable. Mathematically [15], the free energy landscape can be given as, U(nH+ , pH) = U 0 (nH + , pH0 ) − · nH+
CH+ C0H+
(2)
is the free energy released per proton due to the binding at a certain pH with respect to an initial state with energy 0 at pH0 = 7. Writing = − 0 , and measuring the chemical potential in kB T units, we obtain, = ln
CH+ C0H+
(3)
= ln(10) · (pH0 − pH).
(4) n =/
If the number of associated protons at the barrier is at pH0 = 7, it is easy to conclude that by increasing the proton concentration of the medium, the barrier height gets decreased by the amount n =/ , as indicated in Figure 1. With this formalism, the transition rate can be given by the Kramer’s rate [14] of escape of a Brownian particle over a diffusive free energy barrier, as follows, k
−1
1 = D
e well
−U(nH+ )
dnH+
e
U(nH+ )
Dudko–Hummer–Szabo theory [10] to describe the unperturbed free energy surface as, U 0 (nH+ ) =
dnH+ .
(5)
barrier
In order to further calculate the rate analytically, we need to know the unperturbed free energy landscape at a normal pH i.e., at pH0 = 7. We adopt a similar approach as in the
1 G =/
n
H+ n =/
−
1 G =/ −1
n
H+ n =/
1/(1−) (6)
The above potential has the form of a smooth linear cubic barrier for, = 2/3, and a sharp cusp-like barrier for = 1/2. In these two cases, following the formalism adopted in [10], it is trivial to find the analytical expression of the protonation-dependent rate from Eqs. (5) and (6). Thus,
(1)
where the effective free energy at a certain proton concentration or pH (pH = − log CH+ ) is U(nH+ , pH), in kB T units. This effective free energy, U(nH+ , pH) can be obtained by subtracting · nH+ from the energy at pH0 = 7, U 0 (nH+ , pH0 ), (pH0 = − log C0H+ , where C0H+ is the proton concentration at pH0 = 7). Considering a quantitative definition of chemical potential as the free energy released per particle, we can conclude that the above is the amount of free energy released for each protonation process when an external pH is set into the system. Thus, the chemical potential [15,16] at a lower pH (i.e., higher H+ ) lowers the energy barrier of the free energy landscape for a certain folding event. Thus, = 0 + kB T ln
Fig. 1. Schematic diagram (not to scale) of perturbed state free energy landscape U(nH+ , pH) and the intrinsic free energy landscape U 0 (nH+ , pH0 ) are shown in black solid line and dashed gray line respectively with nH+ (t) as the reaction coordinate. n =/ and G =/ are the barrier distance and barrier height of the intrinsic potential U 0 (nH+ , pH0 ) of the protein at pH0 = 7. n =/ is the amount of energy that has been lowered by setting the lower pH.
k(pH) = k0 (pH0 ) 1 −
ln(10) (pH0 − pH)n =/ G =/
× exp G =/
1/−1
ln(10) (pH0 − pH)n =/ 1− 1− G =/
1/ . (7)
Here, G =/ is the barrier height (measured in kB T units) of the folding barrier at the n =/ number of protonation under the fixed initial C0H+ or pH0 . k(pH) is the perturbed rate of the folding process when an external pH is set, and k0 (pH0 ) is the initial rate at pH0 . Thus, assuming that protonation-assisted protein folding is the rate limiting step, we can now quantify the protein free-energy landscape with respect to its intrinsic parameters like k0 (pH0 ), G =/ and n =/ . 3. Results With the above theoretical framework, it is easy to formulate experimentally observable quantities such as the first folding time distribution P(t), and the survival probability S(t) of the folding protein. Under a constant pH, S(t) is found to follow first order kinetics which can be given as, dS(t) = −k(pH) S(t) dt
(8)
S(t) = S0 exp(−k(pH)t)
(9)
P(t) = −
dS(t) = k(pH) S(t). dt
(10)
Replacing k(pH) from Eq. (7) in the above expression gives the pH dependent first passage time distribution P(t) of this folding event. Eq. (7) with Eq. (10) are the key expressions of this study. A Brownian dynamics simulation is run to reproduce the expected experimental behavior of the lag time distribution of the
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D. Chatterjee / Chemical Physics Letters 618 (2015) 94–98
Fig. 2. P(t) vs. t histogram and the globally fitted lagtime distribution using Eq. (7) with Eq. (10) have been shown in the first five subplots. With the extracted parameters that are obtained by fitting the histograms, the k(pH) vs. pH plot shows the pH dependent stability of the protein.
slow protonation-coupled folding event following the overdamped Langevin equation of motion as dnH+ (t) 1 dU(nH+ , pH) =− + (t) ˛ dt dx
(11)
with ˛ = kB T/D where D is the diffusion constant of the each Brownian particle. (t) is the thermal fluctuating force with (t)=0 and (t)(t )=2kB T˛ı(t − t ). Each folding event is signified by the barrier crossing of the Brownian particle in a thermal bath. In this design, the time evolution of the Brownian particle over a linear cubic potential under various fixed pH was monitored with G =/ = 20kB T, n =/ = 3, k0 (pH0 ) = 2.58 × 10−5 s−1 , = 2/3 and pH0 = 7. From this simulation framework, we could construct several histograms of P(t), when the pH values were kept constant at 5.2, 5, 4.8, 4.6 and 4.4 (Figure 2). The histograms of these five different sets with five different pH values were then globally fitted with the analytical expression (Eq. (10)) in combination with Eq. (7). The fitting faithfully reproduces the intrinsic parameters of the free energy landscape of the protein as, G =/ = 18.85 ± 1.575kB T, n =/ = 3.27 ± 1.53, ln k0 (pH0 ) = −11.01 ± 5.03. Using the fitted value of these extracted parameters, k(pH) (Eq. (7)) has been plotted at various pH to visualize the pH dependent protein-stability.
molecule technique may reveal the trend of P(t) or k(pH) at several experimental pH values. The direct fitting of those experimental outputs with the current theoretical expressions such as Eqs. (10) and (7) will enable one to quantify the free energy landscape of the protein under physiological conditions (pH0 = 7) in terms of its various intrinsic parameters like G =/ , n =/ , and k0 (pH0 ). Here, the theoretical expressions Eq. (10) and (7) have been used to fit the histogram obtained from the simulation. The extracted parameters shows a close agreement with the values of the parameters used in the simulation. The pH dependent change in the barrier distance can be found out by the difference between the maxima and minima points of the perturbed energy surface, under several pH, from a starting state, pH0 = 7. It can be easily shown that the pH dependent barrier distance n =/ (pH) will be
n =/ (pH) = n =/
1−
2 ln(10) (pH0 − pH)n =/ 3 G =/
1/2
A plot of Eq. (12) in Figure 3 shows that the barrier distance n =/ (pH) (i.e., the excess number of protonation), decreases with a decrease in pH. As the determination of excess number of protonation n =/ (pH) can be considered as the characteristic molecular control of the folding/unfolding process, this analysis which is
4. Discussion Among many other complex cellular biochemical processes, enzyme catalysis [17] is one of the very important events, where the turn-over rates of the protein in the presence of the enzyme gets increased by several orders. The complex scenario of this enzyme catalyzed reaction has been well studied under Kramer’s theoretical framework, which could correctly reproduce the experimental behavior of the single molecule study [18–20]. The effect of pH for the slow protonation-coupled folding event is found to show a similar behavior. The present study, (see Figure 2), shows that, the dropping of pH from 7 to 4 is capable of increasing the rate of the folding event from 10−5 s−1 to almost 103 s−1 . Thus, once the pH is lowered, biopolymers that are in a standard physiological condition at a pH0 = 7, may undergo a pH triggered conformational change, in a physiologically viable timescale. The measurement of the typical experimental output of an folding event from a well designed single
(12)
Fig. 3. n =/ (pH) vs. pH has been plotted following Eq. (12).
D. Chatterjee / Chemical Physics Letters 618 (2015) 94–98
an application of the analytical tool of the single molecule force spectroscopy, provides a bio-physical method to probe the conformational free energy landscape of the protein by changing pH. In a recent single molecule experimental study, Munoz et al. [2] have observed the slow protonation coupled unfolding of a small ˛-helical BBL protein. On the basis of their observations, they resolved a puzzling fact: even though unfolding of BBL protein at pH = 6 is guided by a unimodal unique conformation, the unfolding process is followed with a bimodal distribution around pH = 7–8. Munoz et al. [2] have argued that at all conditions, BBL protein will have a barrier-less unfolding event, and thus, the unfolding is expected to be associated with a unimodal distribution. However, at around pH = 7–8, the protein will have another slow conformational change to its protonated species and that will be responsible for the bimodal distribution. According to the present theory, the situation can be visualized as a lowering of the conformational barrier between the protonated and unprotonated protein when the pH is decreased from a range of 7–8, to pH = 6, causing the bimodal distribution to converge to an unimodal one. As a result, the protein unfolding at lowered pH is solely governed by the population by the unique minima position of a single conformer on the energy surface. It will be interesting to quantify and analyze this pH dependent conformational energy barrier of BBL protein following the technique discussed here. Thiurmalai et al. [21] have found a theoretical approach based on molecular simulation technique to monitor the pH dependent stability of the protein. Their theory provides a prediction and the experimental validation for the behavior of the protein at different pH even under pulling. A similar approach will be also useful to check how good it is to consider the protonation number as a reaction coordinate during the pH triggered conformational change under a particular set of physical conditions, along the line of the present theoretical prediction. In a minimalistic view, the pH dependent-protein stability can be thought of as a generalized field of research where the stability of the protein gets affected by the presence of activating ions on the conformational free energy pathway. It is important to mention that the approach which has been considered here is sufficiently generalized to capture the essential biophysical details of the problem of ion-affected stability of a protein. In order to extend this present theory for the ion-activated process, with the same present analysis, we can replace the change of chemical potential as, = ln CM+ /C0M+ i.e., = ln(10)(pM0 − pM), (where M+ is any other external ion such as Ca2+ , Mg2+ , H+ etc.; pM = − log(CM + ), and pM 0 = − log(C0M+ ) where C0M+ and CM+ are the concentration of the ion at a standard state and any other concentration perturbed state respectively). The subsequent changes in all the expressions involving (pH0 − pH) with (pM0 − pM) can consider a similar situation of protein stability with the change of any other external ion concentration. Thus, in that case, the number of excess ions bound to the protein with respect to a standard fixed concentration can be considered as n =/ . In a recent experiment, Chu et al. [22] have studied the Mg2+ dependent conformational change of three-helix junction of RNA at the single molecule level by flouroscence correlation spectroscopy (FCS) and FRET methods. By this technique of monitoring the flouroscence correlation of the acceptor and donor dyes attached to two specific sites of the RNA molecule, they could observe Mg2+ dependent rate of several conformational changes of Mg2+ bound RNA. Chu et al. [22] have described their experiment on the basis of cooperative binding of the Mg2+ ions which leads to the conformational change of the protein. The theoretical plots with the numbers of Mg2+ in the range of 1–4 all seemed to show reasonable agreement with the experiments. Chu et al. have mentioned that it is difficult to predict the exact number of the associated Mg2+ as the cooperativity of Mg2+ may not be the same over the entire range of Mg2+ concentration. However, the best fitted plot was obtained when the number of associated Mg2+ was greater than or equal to 4. In
97
Fig. 4. The folding rate k (ms−1 ) vs. pMg of RNA from the experiment of Chu et al. [22] has been plotted (red circles) in the above figure. The fitting of these data with present theory (the blue line with circles) that explains the pH effects on protein conformation, can be applied to extract the essential intrinsic free energy parameters for the Mg2+ dependent stability of the protein. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
order to compare our prediction with the experimental result, we use Chu et al.’s [22] data points for the two state transition of the folding process of Mg2+ bound RNA. We extracted the experimental data points from Fig. (6.B) in Ref.[22] (using a software program (http://plotdigitizer.sourceforge.net)) and plotted the experimental rate vs. Mg2+ in Figure 4. The theoretical expression obtained from the Eq. (13), which has a similar form like Eq. (7) as,
k(pM) = k0 (pM 0 ) 1 −
× exp G =/
ln(10) (pM 0 − pM)n =/ G =/
1/−1
ln(10) (pM 0 − pM)n =/ 1− 1− G =/
1/ . (13)
showed a qualitative agreement with the experimental data and we find that the fitted values at = 2/3 are G =/ = 5.84kB T, n =/ = 1.425, k0 (pM0 ) = 0.00092 ms−1 , considering the standard state as 1 M Mg2+ . As the intrinsic parameters extracted here are dependent on the specification of the standard state we believe that more data points, especially in the range where the rate shows enhancement with the ion concentration, and a better knowledge of the standard state of the protein with respect to the particular ion (Mg2+ ), will help to extract and analyze the parameters, more close to the realistic case. In Eq. (7), = 2/3 or = 1/2 are the special cases when one can analytically extract the rate. At = 1, one can reproduce the rate expression of Bell’s phenomenological formula [23]. It is important to note here that, the form of the potential that has been taken here is generalized and it can be shown that all smooth surfaces can be well represented by a cubic polynomial and the linear cubic potential subsequently can be derived from a smooth combined generalized free energy surface [10]. In addition to that, in a recent study, Hyoun et al.[24] have showed some special cases where the value can be vanishingly small for transition over multiple barriers. Considering a single barrier transition of the protein conformational change, this study has been emphasized with the value of = 2/3 for the sake of simplicity. However, can also be used as a fitting parameter and the value can be determined for a complex protein conformation change.
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D. Chatterjee / Chemical Physics Letters 618 (2015) 94–98
Here, we propose to use the increase in the number of protons as a reaction coordinate to describe the free energy landscape of the protein under a certain pH. Thus, unlike in a single molecule pulling experiment where the conformational change happens due to the external force, here the process of folding/unfolding is chemically activated. In other words, it can be said that even though there will be a change in the internal force of the protein back bone and the associated end-to-end distance, it may not be a good reaction coordinate where the proton transfer is a slow and rate limiting step for the conformational change, favoring the protonation number to be considered as a good reaction coordinate. It is important to note here that, analogous to the pulling experiment where force is a conjugate variable of the reaction co-ordinate (i.e., the end-toend distance), in this case, the chemical potential is a conjugate variable of the reaction coordinate (i.e., the protonation numbers). Thus, considering protonation-coupled activation of the protein as the rate limiting step in the folding/unfolding pathway, we describe the process as an escape over a conformational free energy barrier. The application of this theory has the following implications on the slow protonation-coupled folding/unfolding studies, such as, • The parameters extracted from the above theory can be mapped with a particular state on the folding/unfolding pathway and that will provide the scenario of the process at the molecular level. Moreover, this theory predicts the increase in the number of protons that bind to the protein in response to the change in chemical potential set by the fixed pH of the solvent. The associated number of protons thus helps to identify the critical step in the conformational transition pathway. • In this letter, the model is not system specific and thus can be applied to a large variety of similar systems to extract the free landscape. Further it will be interesting to find out the theoretical framework for a complex process where the mechanical response of biopolymer is probed by an external force under various pH. • An extension of this theoretical concept could inspire experimentalists to investigate similar protonation or any other ion-activated folding/unfolding processes. Moreover, a time
dependent change in pH may reveal other interesting scenarios of the stability of the protein. Acknowledgements The author acknowledges Prof. Olga K. Dudko and Yaojun Zhang for the important discussions and help in learning the computational methods and support from the Center for Theoretical Biological Physics, UCSD. The author is thankful to Prof. Binny J. Cherayil for the constructive comments on the manuscript. References [1] J. Stigler, M. Rief, Proc. Natl. Acad. Sci. U. S. A. 109 (44) (2012) 17814. [2] M. Cerminara, L.A. Campos, R. Ramanathan, V. Muoz, PLOS ONE 8 (10) (2013) e78044. [3] M.E. Bowen, K. Weninger, J. Ernst, S. Chu, A.T. Brunger, Biophys. J. 89 (1) (2005) 690. [4] Q. Huang, R. Opitz, E.-W. Knapp, A. Herrmann, Biophys. J. 82 (2) (2002) 1050. [5] K.A. Merchant, R.B. Best, J.M. Louis, I.V. Gopich, W.A. Eaton, Proc. Natl. Acad. Sci. U. S. A. 104 (5) (2007) 1528. [6] W.J. Greenleaf, M.T. Woodside, S.M. Block, Annu. Rev. Biophys. Biomol. Struct. 36 (1) (2007) 171. [7] P.C. Anthony, C.F. Perez, C. Garca-Garca, S.M. Block, Proc. Natl. Acad. Sci. U. S. A. 109 (5) (2012) 1485. [8] J. Kim, C.-Z. Zhang, X. Zhang, T.A. Springer, Nature 466 (7309) (2010) 992. [9] C. Bustamante, Annu. Rev. Biochem. 77 (1) (2008) 45. [10] O. Dudko, G. Hummer, A. Szabo, Phys. Rev. Lett. 96 (10) (2006). [11] D. Thirumalai, E.P. O’Brien, G. Morrison, C. Hyeon, Annu. Rev. Biophys. 39 (1) (2010) 159. [12] C. Hyeon, D. Thirumalai, J. Phys.: Condens. Matter 19 (11) (2007) 113101. [13] Y. Suzuki, O.K. Dudko, Phys. Rev. Lett. 104 (4) (2010). [14] H.A. Kramers, Physica 7 (4) (1940) 284. [15] R.A. Alberty, Thermodynamics of Biochemical Reactions, Wiley-Interscience, Hoboken, N.J., 2003. [16] R.J. Silbey, R.A. Alberty, Physical Chemistry, Wiley, 2000. [17] X.S. Xie, Science 342 (6165) (2013) 1457. [18] S. Chaudhury, B.J. Cherayil, J. Chem. Phys. 125 (2) (2006) 024904. [19] S. Chaudhury, D. Chatterjee, B.J. Cherayil, J. Chem. Phys. 129 (7) (2008) 075104. [20] B.P. English, et al., Nat. Chem. Biol. 2 (2) (2005) 87. [21] E.P. O’Brien, B.R. Brooks, D. Thirumalai, J. Am. Chem. Soc. 134 (2) (2012) 979. [22] H.D. Kim, G.U. Nienhaus, T. Ha, J.W. Orr, J.R. Williamson, S. Chu, Proc. Natl. Acad. Sci. U. S. A. 99 (7) (2002) 4284. [23] G.I. Bell, Science 200 (4342) (1978) 618. [24] C. Hyeon, D. Thirumalai, J. Chem. Phys. 137 (5) (2012) 055103.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
pH dependent protein stability: A quantitative approach based on Kramer’s barrier escape Debarati Chatterjee Center for Theoretical Biological Physics, University of California, La Jolla, San Diego, CA 92093, USA
a r t i c l e
i n f o
Article history: Received 27 July 2014 In final form 3 November 2014 Available online 8 November 2014
a b s t r a c t The folding–unfolding mechanism of a protein in a living cell is the result of its molecular response to various perturbations due to changes in the chemical environment such as pH, ion concentration, ligand binding and many other complex chemical reactions. In this letter, a theoretical scheme based on single molecule force spectroscopy experiments and related theories, has been followed to capture the effect of the change of pH for a slow protonation-coupled folding–unfolding process, and the free energy landscape has been quantified in terms of protonation numbers, barrier height and pH, following Kramer’s barrier crossing formalism. © 2014 Elsevier B.V. All rights reserved.
1. Introduction In a living cell, the mechanical response and folding–unfolding mechanism of a biopolymer is often an outcome of a series of chemical reactions. Thus a change in the folding kinetics due to the influence of any other chemical agent or ions is an important aspect of the folding mechanism. Many of the protein folding–unfolding kinetics are found to be triggered by Ca2+ , H+ or other ion concentrations [1–4]. These ions are found to play an important role in maintaining a well regulated physiologically viable timescale of the cell machinery by modulating the rate of the folding–unfolding kinetics of the involved proteins. Single molecule experiments based on the FRET (flouroscence resonance energy transfer) have also monitored the folding–unfolding process when they are triggered by chemical denaturants [5]. Thus, how the surrounding ions can change the stability of the protein is a thought provoking area of study in the field of biophysical research. Recent developments in various single molecule force spectroscopy techniques have enabled us to acquire a quantitative molecular view of the protein folding kinetics [6–9]. In this method, a biopolymer is probed by an externally applied force and the mechanical response of the biopolymer is monitored. The intrinsic free energy landscape of the biopolymer can be extracted in terms of its end-to-end distance from the related theoretical framework [10,11]. However, the effect of chemical activation on the folding unfolding process in still not satisfactorily explained in detail. Moreover, in a complex protein folding scenario, often the force spectroscopic data shows some
E-mail address: [email protected] http://dx.doi.org/10.1016/j.cplett.2014.11.002 0009-2614/© 2014 Elsevier B.V. All rights reserved.
unusual behavior. This is due to the involvement of other slower variables rather than end-to-end distance among the available vast number of degrees of freedom, originating from the biomolecule and its surroundings. In that case, only the incorporation of the proper reaction coordinate along the rate limiting variable, seems to provide an adequate description of the overall process [12,13]. In this letter, we highlight a folding scenario where the folding–unfolding event of the protein is pH dependent and we assume here that the process can be described by a slow protonation-coupled folding pathway. In the case of this slow protonation assisted protein folding, we propose here that the effect of the rate enhancement of the unfolding process by pH may be a direct effect of activated conformational change of the protein machinery due to an increase in [H+ ] concentration. Moreover, this pH-induced rapid conformational change of protein may resemble the scenario where the unfolding rate of a protein increases due to an external pulling force in a single molecule pulling experiment. The Dudko–Hummer–Szabo theory [10] provides an analytical method and an exact result based on the concept of Kramers’ escape [14] over a single barrier to quantify the activation barrier and the transition distance, from the unfolding rate vs. force plot obtained from experiments. The theory considers the unfolding process under an applied external force as a single barrier crossing process with respect to the end-to-end distance of the protein as the reaction coordinate. We believe, that a similar approach should be applicable here to extract the various quantities of interest from the pH dependent folding–unfolding experiment. Hence, on the basis of the above concept, considering the protonation number as the reaction coordinate, we aim to deduce a general theoretical background for the process of conformational change and provide a method to
D. Chatterjee / Chemical Physics Letters 618 (2015) 94–98
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quantify the system in terms of its intrinsic parameters like rate of the process, barrier height and barrier distance, at the usual physiological condition, pH0 = 7. Thus, finding and quantifying the hidden free energy landscape of the unfolding event by simply probing the protein with pH via single molecule study is the main aim of this letter. 2. Theoretical background Let us consider a situation when, in the presence of an external pH, the folding rate of a protein increases compared to the folding rate of the same protein at a normal cellular pH value of pH0 = 7. In order to visualize the effect of the externally applied pH, we assume here that protonations activate the protein machinery on the folding pathway and we describe this overall transition of proton-assisted folding of the protein as a thermally activated single barrier escape problem. When the pH is set to be lower than pH0 = 7, it helps to lower the barrier of the folding pathway and the rate becomes faster. Thus, here, H+ ions trigger the conformational change in the protein. We formulate the free energy landscape U, where we choose the reaction coordinate as the number of added extra protons nH+ (t) (with respect to the average number of protons associated to the protein at pH0 = 7) getting bound to the proteins at any instant to trigger the conformational change at a certain pH under a constant temperature. This facilitates the transition from the unfolded to the final folded state. nH+ (t) at any instant, is a thermally fluctuating quantity, and is a random variable. Mathematically [15], the free energy landscape can be given as, U(nH+ , pH) = U 0 (nH + , pH0 ) − · nH+
CH+ C0H+
(2)
is the free energy released per proton due to the binding at a certain pH with respect to an initial state with energy 0 at pH0 = 7. Writing = − 0 , and measuring the chemical potential in kB T units, we obtain, = ln
CH+ C0H+
(3)
= ln(10) · (pH0 − pH).
(4) n =/
If the number of associated protons at the barrier is at pH0 = 7, it is easy to conclude that by increasing the proton concentration of the medium, the barrier height gets decreased by the amount n =/ , as indicated in Figure 1. With this formalism, the transition rate can be given by the Kramer’s rate [14] of escape of a Brownian particle over a diffusive free energy barrier, as follows, k
−1
1 = D
e well
−U(nH+ )
dnH+
e
U(nH+ )
Dudko–Hummer–Szabo theory [10] to describe the unperturbed free energy surface as, U 0 (nH+ ) =
dnH+ .
(5)
barrier
In order to further calculate the rate analytically, we need to know the unperturbed free energy landscape at a normal pH i.e., at pH0 = 7. We adopt a similar approach as in the
1 G =/
n
H+ n =/
−
1 G =/ −1
n
H+ n =/
1/(1−) (6)
The above potential has the form of a smooth linear cubic barrier for, = 2/3, and a sharp cusp-like barrier for = 1/2. In these two cases, following the formalism adopted in [10], it is trivial to find the analytical expression of the protonation-dependent rate from Eqs. (5) and (6). Thus,
(1)
where the effective free energy at a certain proton concentration or pH (pH = − log CH+ ) is U(nH+ , pH), in kB T units. This effective free energy, U(nH+ , pH) can be obtained by subtracting · nH+ from the energy at pH0 = 7, U 0 (nH+ , pH0 ), (pH0 = − log C0H+ , where C0H+ is the proton concentration at pH0 = 7). Considering a quantitative definition of chemical potential as the free energy released per particle, we can conclude that the above is the amount of free energy released for each protonation process when an external pH is set into the system. Thus, the chemical potential [15,16] at a lower pH (i.e., higher H+ ) lowers the energy barrier of the free energy landscape for a certain folding event. Thus, = 0 + kB T ln
Fig. 1. Schematic diagram (not to scale) of perturbed state free energy landscape U(nH+ , pH) and the intrinsic free energy landscape U 0 (nH+ , pH0 ) are shown in black solid line and dashed gray line respectively with nH+ (t) as the reaction coordinate. n =/ and G =/ are the barrier distance and barrier height of the intrinsic potential U 0 (nH+ , pH0 ) of the protein at pH0 = 7. n =/ is the amount of energy that has been lowered by setting the lower pH.
k(pH) = k0 (pH0 ) 1 −
ln(10) (pH0 − pH)n =/ G =/
× exp G =/
1/−1
ln(10) (pH0 − pH)n =/ 1− 1− G =/
1/ . (7)
Here, G =/ is the barrier height (measured in kB T units) of the folding barrier at the n =/ number of protonation under the fixed initial C0H+ or pH0 . k(pH) is the perturbed rate of the folding process when an external pH is set, and k0 (pH0 ) is the initial rate at pH0 . Thus, assuming that protonation-assisted protein folding is the rate limiting step, we can now quantify the protein free-energy landscape with respect to its intrinsic parameters like k0 (pH0 ), G =/ and n =/ . 3. Results With the above theoretical framework, it is easy to formulate experimentally observable quantities such as the first folding time distribution P(t), and the survival probability S(t) of the folding protein. Under a constant pH, S(t) is found to follow first order kinetics which can be given as, dS(t) = −k(pH) S(t) dt
(8)
S(t) = S0 exp(−k(pH)t)
(9)
P(t) = −
dS(t) = k(pH) S(t). dt
(10)
Replacing k(pH) from Eq. (7) in the above expression gives the pH dependent first passage time distribution P(t) of this folding event. Eq. (7) with Eq. (10) are the key expressions of this study. A Brownian dynamics simulation is run to reproduce the expected experimental behavior of the lag time distribution of the
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D. Chatterjee / Chemical Physics Letters 618 (2015) 94–98
Fig. 2. P(t) vs. t histogram and the globally fitted lagtime distribution using Eq. (7) with Eq. (10) have been shown in the first five subplots. With the extracted parameters that are obtained by fitting the histograms, the k(pH) vs. pH plot shows the pH dependent stability of the protein.
slow protonation-coupled folding event following the overdamped Langevin equation of motion as dnH+ (t) 1 dU(nH+ , pH) =− + (t) ˛ dt dx
(11)
with ˛ = kB T/D where D is the diffusion constant of the each Brownian particle. (t) is the thermal fluctuating force with (t)=0 and (t)(t )=2kB T˛ı(t − t ). Each folding event is signified by the barrier crossing of the Brownian particle in a thermal bath. In this design, the time evolution of the Brownian particle over a linear cubic potential under various fixed pH was monitored with G =/ = 20kB T, n =/ = 3, k0 (pH0 ) = 2.58 × 10−5 s−1 , = 2/3 and pH0 = 7. From this simulation framework, we could construct several histograms of P(t), when the pH values were kept constant at 5.2, 5, 4.8, 4.6 and 4.4 (Figure 2). The histograms of these five different sets with five different pH values were then globally fitted with the analytical expression (Eq. (10)) in combination with Eq. (7). The fitting faithfully reproduces the intrinsic parameters of the free energy landscape of the protein as, G =/ = 18.85 ± 1.575kB T, n =/ = 3.27 ± 1.53, ln k0 (pH0 ) = −11.01 ± 5.03. Using the fitted value of these extracted parameters, k(pH) (Eq. (7)) has been plotted at various pH to visualize the pH dependent protein-stability.
molecule technique may reveal the trend of P(t) or k(pH) at several experimental pH values. The direct fitting of those experimental outputs with the current theoretical expressions such as Eqs. (10) and (7) will enable one to quantify the free energy landscape of the protein under physiological conditions (pH0 = 7) in terms of its various intrinsic parameters like G =/ , n =/ , and k0 (pH0 ). Here, the theoretical expressions Eq. (10) and (7) have been used to fit the histogram obtained from the simulation. The extracted parameters shows a close agreement with the values of the parameters used in the simulation. The pH dependent change in the barrier distance can be found out by the difference between the maxima and minima points of the perturbed energy surface, under several pH, from a starting state, pH0 = 7. It can be easily shown that the pH dependent barrier distance n =/ (pH) will be
n =/ (pH) = n =/
1−
2 ln(10) (pH0 − pH)n =/ 3 G =/
1/2
A plot of Eq. (12) in Figure 3 shows that the barrier distance n =/ (pH) (i.e., the excess number of protonation), decreases with a decrease in pH. As the determination of excess number of protonation n =/ (pH) can be considered as the characteristic molecular control of the folding/unfolding process, this analysis which is
4. Discussion Among many other complex cellular biochemical processes, enzyme catalysis [17] is one of the very important events, where the turn-over rates of the protein in the presence of the enzyme gets increased by several orders. The complex scenario of this enzyme catalyzed reaction has been well studied under Kramer’s theoretical framework, which could correctly reproduce the experimental behavior of the single molecule study [18–20]. The effect of pH for the slow protonation-coupled folding event is found to show a similar behavior. The present study, (see Figure 2), shows that, the dropping of pH from 7 to 4 is capable of increasing the rate of the folding event from 10−5 s−1 to almost 103 s−1 . Thus, once the pH is lowered, biopolymers that are in a standard physiological condition at a pH0 = 7, may undergo a pH triggered conformational change, in a physiologically viable timescale. The measurement of the typical experimental output of an folding event from a well designed single
(12)
Fig. 3. n =/ (pH) vs. pH has been plotted following Eq. (12).
D. Chatterjee / Chemical Physics Letters 618 (2015) 94–98
an application of the analytical tool of the single molecule force spectroscopy, provides a bio-physical method to probe the conformational free energy landscape of the protein by changing pH. In a recent single molecule experimental study, Munoz et al. [2] have observed the slow protonation coupled unfolding of a small ˛-helical BBL protein. On the basis of their observations, they resolved a puzzling fact: even though unfolding of BBL protein at pH = 6 is guided by a unimodal unique conformation, the unfolding process is followed with a bimodal distribution around pH = 7–8. Munoz et al. [2] have argued that at all conditions, BBL protein will have a barrier-less unfolding event, and thus, the unfolding is expected to be associated with a unimodal distribution. However, at around pH = 7–8, the protein will have another slow conformational change to its protonated species and that will be responsible for the bimodal distribution. According to the present theory, the situation can be visualized as a lowering of the conformational barrier between the protonated and unprotonated protein when the pH is decreased from a range of 7–8, to pH = 6, causing the bimodal distribution to converge to an unimodal one. As a result, the protein unfolding at lowered pH is solely governed by the population by the unique minima position of a single conformer on the energy surface. It will be interesting to quantify and analyze this pH dependent conformational energy barrier of BBL protein following the technique discussed here. Thiurmalai et al. [21] have found a theoretical approach based on molecular simulation technique to monitor the pH dependent stability of the protein. Their theory provides a prediction and the experimental validation for the behavior of the protein at different pH even under pulling. A similar approach will be also useful to check how good it is to consider the protonation number as a reaction coordinate during the pH triggered conformational change under a particular set of physical conditions, along the line of the present theoretical prediction. In a minimalistic view, the pH dependent-protein stability can be thought of as a generalized field of research where the stability of the protein gets affected by the presence of activating ions on the conformational free energy pathway. It is important to mention that the approach which has been considered here is sufficiently generalized to capture the essential biophysical details of the problem of ion-affected stability of a protein. In order to extend this present theory for the ion-activated process, with the same present analysis, we can replace the change of chemical potential as, = ln CM+ /C0M+ i.e., = ln(10)(pM0 − pM), (where M+ is any other external ion such as Ca2+ , Mg2+ , H+ etc.; pM = − log(CM + ), and pM 0 = − log(C0M+ ) where C0M+ and CM+ are the concentration of the ion at a standard state and any other concentration perturbed state respectively). The subsequent changes in all the expressions involving (pH0 − pH) with (pM0 − pM) can consider a similar situation of protein stability with the change of any other external ion concentration. Thus, in that case, the number of excess ions bound to the protein with respect to a standard fixed concentration can be considered as n =/ . In a recent experiment, Chu et al. [22] have studied the Mg2+ dependent conformational change of three-helix junction of RNA at the single molecule level by flouroscence correlation spectroscopy (FCS) and FRET methods. By this technique of monitoring the flouroscence correlation of the acceptor and donor dyes attached to two specific sites of the RNA molecule, they could observe Mg2+ dependent rate of several conformational changes of Mg2+ bound RNA. Chu et al. [22] have described their experiment on the basis of cooperative binding of the Mg2+ ions which leads to the conformational change of the protein. The theoretical plots with the numbers of Mg2+ in the range of 1–4 all seemed to show reasonable agreement with the experiments. Chu et al. have mentioned that it is difficult to predict the exact number of the associated Mg2+ as the cooperativity of Mg2+ may not be the same over the entire range of Mg2+ concentration. However, the best fitted plot was obtained when the number of associated Mg2+ was greater than or equal to 4. In
97
Fig. 4. The folding rate k (ms−1 ) vs. pMg of RNA from the experiment of Chu et al. [22] has been plotted (red circles) in the above figure. The fitting of these data with present theory (the blue line with circles) that explains the pH effects on protein conformation, can be applied to extract the essential intrinsic free energy parameters for the Mg2+ dependent stability of the protein. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
order to compare our prediction with the experimental result, we use Chu et al.’s [22] data points for the two state transition of the folding process of Mg2+ bound RNA. We extracted the experimental data points from Fig. (6.B) in Ref.[22] (using a software program (http://plotdigitizer.sourceforge.net)) and plotted the experimental rate vs. Mg2+ in Figure 4. The theoretical expression obtained from the Eq. (13), which has a similar form like Eq. (7) as,
k(pM) = k0 (pM 0 ) 1 −
× exp G =/
ln(10) (pM 0 − pM)n =/ G =/
1/−1
ln(10) (pM 0 − pM)n =/ 1− 1− G =/
1/ . (13)
showed a qualitative agreement with the experimental data and we find that the fitted values at = 2/3 are G =/ = 5.84kB T, n =/ = 1.425, k0 (pM0 ) = 0.00092 ms−1 , considering the standard state as 1 M Mg2+ . As the intrinsic parameters extracted here are dependent on the specification of the standard state we believe that more data points, especially in the range where the rate shows enhancement with the ion concentration, and a better knowledge of the standard state of the protein with respect to the particular ion (Mg2+ ), will help to extract and analyze the parameters, more close to the realistic case. In Eq. (7), = 2/3 or = 1/2 are the special cases when one can analytically extract the rate. At = 1, one can reproduce the rate expression of Bell’s phenomenological formula [23]. It is important to note here that, the form of the potential that has been taken here is generalized and it can be shown that all smooth surfaces can be well represented by a cubic polynomial and the linear cubic potential subsequently can be derived from a smooth combined generalized free energy surface [10]. In addition to that, in a recent study, Hyoun et al.[24] have showed some special cases where the value can be vanishingly small for transition over multiple barriers. Considering a single barrier transition of the protein conformational change, this study has been emphasized with the value of = 2/3 for the sake of simplicity. However, can also be used as a fitting parameter and the value can be determined for a complex protein conformation change.
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D. Chatterjee / Chemical Physics Letters 618 (2015) 94–98
Here, we propose to use the increase in the number of protons as a reaction coordinate to describe the free energy landscape of the protein under a certain pH. Thus, unlike in a single molecule pulling experiment where the conformational change happens due to the external force, here the process of folding/unfolding is chemically activated. In other words, it can be said that even though there will be a change in the internal force of the protein back bone and the associated end-to-end distance, it may not be a good reaction coordinate where the proton transfer is a slow and rate limiting step for the conformational change, favoring the protonation number to be considered as a good reaction coordinate. It is important to note here that, analogous to the pulling experiment where force is a conjugate variable of the reaction co-ordinate (i.e., the end-toend distance), in this case, the chemical potential is a conjugate variable of the reaction coordinate (i.e., the protonation numbers). Thus, considering protonation-coupled activation of the protein as the rate limiting step in the folding/unfolding pathway, we describe the process as an escape over a conformational free energy barrier. The application of this theory has the following implications on the slow protonation-coupled folding/unfolding studies, such as, • The parameters extracted from the above theory can be mapped with a particular state on the folding/unfolding pathway and that will provide the scenario of the process at the molecular level. Moreover, this theory predicts the increase in the number of protons that bind to the protein in response to the change in chemical potential set by the fixed pH of the solvent. The associated number of protons thus helps to identify the critical step in the conformational transition pathway. • In this letter, the model is not system specific and thus can be applied to a large variety of similar systems to extract the free landscape. Further it will be interesting to find out the theoretical framework for a complex process where the mechanical response of biopolymer is probed by an external force under various pH. • An extension of this theoretical concept could inspire experimentalists to investigate similar protonation or any other ion-activated folding/unfolding processes. Moreover, a time
dependent change in pH may reveal other interesting scenarios of the stability of the protein. Acknowledgements The author acknowledges Prof. Olga K. Dudko and Yaojun Zhang for the important discussions and help in learning the computational methods and support from the Center for Theoretical Biological Physics, UCSD. The author is thankful to Prof. Binny J. Cherayil for the constructive comments on the manuscript. References [1] J. Stigler, M. Rief, Proc. Natl. Acad. Sci. U. S. A. 109 (44) (2012) 17814. [2] M. Cerminara, L.A. Campos, R. Ramanathan, V. Muoz, PLOS ONE 8 (10) (2013) e78044. [3] M.E. Bowen, K. Weninger, J. Ernst, S. Chu, A.T. Brunger, Biophys. J. 89 (1) (2005) 690. [4] Q. Huang, R. Opitz, E.-W. Knapp, A. Herrmann, Biophys. J. 82 (2) (2002) 1050. [5] K.A. Merchant, R.B. Best, J.M. Louis, I.V. Gopich, W.A. Eaton, Proc. Natl. Acad. Sci. U. S. A. 104 (5) (2007) 1528. [6] W.J. Greenleaf, M.T. Woodside, S.M. Block, Annu. Rev. Biophys. Biomol. Struct. 36 (1) (2007) 171. [7] P.C. Anthony, C.F. Perez, C. Garca-Garca, S.M. Block, Proc. Natl. Acad. Sci. U. S. A. 109 (5) (2012) 1485. [8] J. Kim, C.-Z. Zhang, X. Zhang, T.A. Springer, Nature 466 (7309) (2010) 992. [9] C. Bustamante, Annu. Rev. Biochem. 77 (1) (2008) 45. [10] O. Dudko, G. Hummer, A. Szabo, Phys. Rev. Lett. 96 (10) (2006). [11] D. Thirumalai, E.P. O’Brien, G. Morrison, C. Hyeon, Annu. Rev. Biophys. 39 (1) (2010) 159. [12] C. Hyeon, D. Thirumalai, J. Phys.: Condens. Matter 19 (11) (2007) 113101. [13] Y. Suzuki, O.K. Dudko, Phys. Rev. Lett. 104 (4) (2010). [14] H.A. Kramers, Physica 7 (4) (1940) 284. [15] R.A. Alberty, Thermodynamics of Biochemical Reactions, Wiley-Interscience, Hoboken, N.J., 2003. [16] R.J. Silbey, R.A. Alberty, Physical Chemistry, Wiley, 2000. [17] X.S. Xie, Science 342 (6165) (2013) 1457. [18] S. Chaudhury, B.J. Cherayil, J. Chem. Phys. 125 (2) (2006) 024904. [19] S. Chaudhury, D. Chatterjee, B.J. Cherayil, J. Chem. Phys. 129 (7) (2008) 075104. [20] B.P. English, et al., Nat. Chem. Biol. 2 (2) (2005) 87. [21] E.P. O’Brien, B.R. Brooks, D. Thirumalai, J. Am. Chem. Soc. 134 (2) (2012) 979. [22] H.D. Kim, G.U. Nienhaus, T. Ha, J.W. Orr, J.R. Williamson, S. Chu, Proc. Natl. Acad. Sci. U. S. A. 99 (7) (2002) 4284. [23] G.I. Bell, Science 200 (4342) (1978) 618. [24] C. Hyeon, D. Thirumalai, J. Chem. Phys. 137 (5) (2012) 055103.