Modelling the Unfolding Pathway of Biomolecules: Theoretical Approach and Experimental Prospect

Modelling the unfolding pathway of biomolecules: theoretical approach and experimental prospect Carlos A. Plata and Antonio Prados Abstract We analyse the unfolding pathway of biomolecules comprising several independent modules in pulling experiments. In a recently proposed model, a critical velocity vchas been predicted, such that for pulling speeds v>vcit is the module at the pulled end that opens first, whereas for v<vcit is the weakest. Here, we introduce a variant of the model that is closer to the experimental setup, and discuss the robustness of the emergence of the critical velocity and of its dependence on the model parameters. We also propose a possible experiment to test the theoretical predictions of the model, which seems feasible with state-of-art molecular engineering techniques. Key words: Single-molecule experiments, free energy, unfolding pathway, critical velocity, force-extension curve, modular proteins 1 Introduction The development of the so-called single-molecule experiments in the last decades has made it possible to carry out research at the molecular level. Biophysics is, undoubtedly, one of the fields where these techniques have had a bigger impact, triggering a whole new area of investigation on the elastic properties of biomolecules. Recent accounts of the current development of this enticing field can be found in Refs. [1–4]. Carlos A. Plata F´ ısica Te´ orica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Seville, Spain, EU, e-mail: [email protected] Antonio Prados F´ ısica Te´ orica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Seville, Spain, EU, e-mail: [email protected] 1 arXiv:1706.02475v1 [cond-mat.soft] 8 Jun 2017 2 Carlos A. Plata and Antonio Prados Atomic force microscopy (AFM) stands out because of its extensive use. In particular, the role played by AFM in the study of modular proteins is crucial [5–7]. Figure 1 shows a sketch of the experimental setup in a pulling experiment of a molecule comprising several modules. The biomolecule is stretched between the platform and the tip of the cantilever. The spring constant of the cantilever is kc, which is usually in the range of 100 pN/nm. Here, we consider the simplest situation, in which the total length of the system ∆Zp=∆Zc+Xis controlled. The stretching of the molecule makes the cantilever bend by ∆Zc, and then the force can be recorded as F=kc∆Zc. The outcome of the above described experiment is a force-extension curve, similar to panel b) in Fig. 1. This force-extension curve provides a fingerprint of the elastomechanical properties of the molecule under study. When molecules composed of several structural units, such as modular polyproteins, are pulled, a sawtooth pattern comes about in the force-extension curve [5–7]. At certain values of the length, there are almost vertical “force rips”: each force rip marks the unfolding of one module. Interestingly, these force rips already appear when the molecule is quasistatically pulled, a limit that can be explained by means of an equilibrium statistical mechanics description [8]. When the molecule is pulled at a finite rate, the appearance of these force rips can still be explained by the system partially sweeping the metastable region of the equilibrium branches [9]. The unfolding pathway is, roughly, the order and the way in which the structural units of the system unravel. It has been recently found out that the unfolding pathway depends on the pulling velocity and the pulling direction [7, 10–12]. Particularly, in [11], different unfolding pathways are observed in SMD simulations on the Maltose Binding Protein. The authors reported that for low pulling speeds the first unit to unfold is the least stable, whereas for high pulling speed it is the closest to the pulled end, regardless of their relative stability. Very recently, a toy-like model has been proposed to qualitatively understand the above experimental framework [13]. Specifically, each module is represented by a nonlinear spring, characterised by a bistable free energy that depends on the module extension. Therein, the two basins represent its folded and unfolded states. The spatial structure of the system is retained in its simplest way: each module extends from the end point of the previous one to its own endpoint (which coincides with the start point of the next). Moreover, each module endpoint obeys an overdamped Langevin equation with forces stemming from the bistable free energies and white noise forces with amplitudes verifying the fluctuation-dissipation theorem. In the above model, the unfolding pathway was found to depend on the pulling velocity. In the simplest non-trivial case, there is only one module that is different from the rest, which is also the furthest from the pulled end. In this situation, only one critical velocity vcshows up: for pulling velocities vp>vc, it is the weakest module that opens first but for vp>vcit is the module at the pulled end. In addition, analytical results were derived for this critical velocity by introducing some approximations: mainly two, (i) perfect length control and (ii) the deterministic approximation, that is, our neglecting of the stochastic forces. This was done by means of a perturbative solution of the deterministic equations in both the pulling velocity Modelling the unfolding pathway of biomolecules 3 Fig. 1 a: Sketch of the experimental setup in an AFM experiment with a modular protein. The position of the platform is shifted ∆Zpfrom 1 to 2, producing an elongation of Xover the molecule and bending the cantilever a magnitude ∆Zc. From 2 to 3 the force is almost relaxed because of the unravelling of one of the modules. b: Force-extension curve in a typical lenght-control AFM experiment with a polyprotein. Each rip in the force accounts for the unfolding of a module. Taken from [3]. and an asymmetry parameter, which measures how different the potentials of the modules are. The main aim of this work is twofold. First, we would like to refine the above theoretical framework, making it closer to the experimental setup in AFM. In particular, we would like to look into the effect of a more realistic modelling of the length-control device. Instead of considering perfect length-control, we consider a device with a finite value of the stiffness, both at the end of the one-dimensional chain (as originally depicted in Ref. [13]) and at the start point thereof, which is where it is usually situated in the AFM experiments, see Fig. 1. Second, we would 4 Carlos A. Plata and Antonio Prados like to discuss how our theory could be checked in a real experiment with modular proteins. This chapter is structured as follows. First, we introduce the original model and discuss its most relevant results in section 2. In section 3, we study the role played by the location of the restoring spring and the finite value of its spring constant. We provide some details about the free energy modelling employed for each of the modules in section 4. Section 5 is devoted to discuss a possible AFM experiment in order to test our theory. Finally, we wrap up the main conclusions which emerge from our work. 2 Revision of the model and previous results Here, we briefly review the model that was originally put forward in Ref. [13]. We consider a polyprotein comprising Nmodules. When the molecule is stretched, the simplest description is to portray it as a one-dimensional chain. We define the coordinates qi,(i=0,...,N)in such a way that the i-th unit extends from qi−1to qi, the extension of the i-th unit is xi=qi−qi−1. Moreover, as shown in Fig. 2, the pulling device is assumed to be connected to the right (pulled) end of the chain. Fig. 2 Sketch of the model for a molecule comprising four units. Therein, the units are denoted by rectangles and have potentials Ui(xi), with xibeing the extension of the i-th unit’s. The unit endpoints qiare represented by the beads, and the extensions are thus xi=qi−qi−1(by definition, q0=0). The spring stands for the length-controlling device attached to the pulled end q4, whose contribution to the system free energy is assumed to be harmonic with stiffness kc. We assume Langevin dynamics for the qicoordinates (q0=0), γ˙qi=−∂ ∂qi A(q0,...,qN)+ζii>0,(1) in which the ζiare Gaussian white noise forces, such that hζi(t)i=0,hζi(t)ζj(t0)i=2γkBTδi jδ(t−t0),(2) with kBbeing the Boltzmann constant, and γand Tbeing the friction coefficient (assumed to be common for all the units) and the temperature of the fluid in which the protein is immersed, respectively. The global free energy function of the system Modelling the unfolding pathway of biomolecules 5 is1 A(q0,...,qN) = N ∑ i=1 Ui(qi−qi−1) + 1 2kc(L−qN)2.(3) In the previous equation, we have considered an elastic term due to the finite stiffness kcof the controlling device, which is located at the pulling end as shown in Fig. 2. Finally, Lstands for the desired length program and Ui(xi)is the single unit contribution, which only depends on the extension, to A. Consequently, the force exerted over the biomolecule is kc(L−qN). We consider length-controlled experiments at constant pulling velocity, that is, ˙ L=vp. When kc→∞, the control over the length is perfect and qN→Lin such a way that kc(L−qN)→F, being Fa Lagrange multiplier. That is, the perfect length-controlled situation is the same that a force-controlled one but with Fthe force needed to maintain a total length equal to L. Note that there is no contribution to the free energy coming from the pulling device in this limit, since kc(L−qN)2/2=F2/(2kc)→0. The approach in Ref. [13] tries to keep things as simple as possible. Then, the evolution equations for the extensions are written by assuming (i) perfect length control and (ii) the deterministic approximation, obtained by neglecting the noise terms. Note that the evolution equations in the latter approximation are sometimes called the macroscopic equations [14], which are γ˙x1=−U0 1(x1)+U0 2(x2),(4a) γ˙xi=−2U0 i(xi)+U0 i+1(xi+1)+U0 i−1(xi−1),1<i<N,(4b) γ˙xN=−2U0 N(xN)+U0 N−1(xN−1)+F,(4c) F=γvp+U0 N(xN).(4d) So far, nothing has been said about the shape of the single-unit contributions Ui to the free energy. In order to maintain a general approach, we only request the functions Ui(x)−Fx to become a double well for some interval of forces. Each well stands for the folded and the unfolded basins of each module. Now, we separate these functions in a main part common to all units and a separation from this main part weighted by an asymmetry parameter ξ, Ui(x) = U(x) + ξδUi(x),U0 i(x) = U0(x)+ξδ fi(x).(5) We have done the same separation in the derivative, by defining δfi(x) = δU0 i(x). It is possible to solve the system (4) by means of a perturbative expansion in the pulling velocity vpand the asymmetry ξ. Indeed, if we retain only linear order terms in vpand ξ, the corrections due to finite pulling rate and asymmetry are not coupled. This perturbative solution, when the system starts from an initial condition in which all the units are folded, is shown to be [13] 1In Ref. [13], this free energy was denoted by G. Here, we have preferred to employ Abecause the relevant potential in length-controlled situations is the Hemlholtz-like free energy, and Gis usually the notation reserved for the Gibbs-like potential G=A−FL, which is the relevant one in force-controlled experiments [8,9]. 6 Carlos A. Plata and Antonio Prados xi=`+ ξδ f−vpγN2−1 6N U00(`)+ vpγi(i−1) 2N−ξδ fi(`) U00(`),(6) where `=L/Nstands for the specific length per module and the over-bar means average over the units. Note that, if all the free energies are equal (ξ=0) and we are not pulling (vp=0), the total length will be reasonably equally distributed among all the units. Moreover, it is worth emphasising that this solution is approximate, it diverges when U00(`)→0. This shows that this perturbative solution breaks down when the average length per module `reaches the stability threshold `b, such that U00(`b) = 0. We are interested in a criterion that allows us to discern which unit is the first to unfold and we hope that our perturbative solution is good enough in this regard. Since the folded state ceases to exist when xreaches `b, it is reasonable to assume that the first module to unfold is precisely the one for which xi=`bis attained for the shortest time. In Eq. (6), we can see that the finite pulling term favours the unfolding of modules that are nearer to the pulled end, whereas the asymmetry term favours the unfolding of the weaker units (those with the lowest values of δfi). We can compute the pulling velocities vi(j)for which each couple of modules (i,j),j>i, reach simultaneously the stability threshold. They are determined by the condition xi(`c) = xj(`c) = `b,(7) which gives both the value of `c(or time tc) at which the stability threshold is reached and the relationship between vpand ξ. Then, in a specific system with known δfi’s we can predict what are the critical velocities that separate regions inside which the first unit to unfold is different. In Ref. [13], some examples of the use of this theory are provided, which show a good agreement with simulations of the Langevin dynamics (1). The simplest configuration in which a critical velocity arises is the following. Let us consider a chain of Nunits, all of them with the same contribution to the free energy, except the first one (the furthest to the pulled end). Therefore, δfi(x) = 0, i6=1, δf1(x)6=0. Moreover we will assume that δf1(`b)<0, that is, the first module is weaker than the rest. For such a configuration, there appears only one critical velocity, which is given by [13] γvc ξ=−δf1(`b) N−1.(8) For vp<vc, the first module to unfold is the weakest one, whereas for vp>vc the unfolding starts from the pulled end. In Ref. [13], a more general situation is investigated but, here, we restrict ourselves to this configuration. Modelling the unfolding pathway of biomolecules 7 3 Relevance of the stiffness In a real AFM experiment, the stiffness is finite and, as a result, the control over the length is not perfect. Furthermore, the position that is externally controlled is, usually, that of the platform and the main elastic force stems from the bending of the tip of the cantilever, as depicted in Fig. 1. Thus, it seems more reasonable to model the pulling of the biomolecule in the way sketched in Fig. 3. Fig. 3 Sketch of the model for a protein with four units. It is identical toFig. 2, except for the position of the length-controlling device, which is now located at the fixed end. Some authors [15] have used other elastic reactions that reflect the attachment by means of flexible linkers among the platform and the pulled end qN, and between consecutive modules. Here, we will consider a perfect absorption, in order to keep the model as simple as possible. In the next two subsections, we study the effect of the finite value of the stiffness kcand the location of the spring, respectively, on the unfolding pathway. 3.1 Finite stiffness Here, we still consider the model depicted in Fig. 2, that is, the spring is situated at the end of the chain, but with a finite value of the stiffness kc. Also, we consider the macroscopic equations (zero noise), which are γ˙x1=−U0 1(x1)+U0 2(x2),(9a) γ˙xi=−2U0 i(xi)+U0 i+1(xi+1)+U0 i−1(xi−1),1<i<N,(9b) γ˙xN=−2U0 N(xN)+U0 N−1(xN−1)+kc L− N ∑ k=1 xk!.(9c) This system differs from that in Eq. (4) because in the last equation the Lagrange multiplier Fis substituted by the harmonic force kc(L−∑N k=1xk). As in the previous case, this system is analytically solvable by means of a perturbative expansion in vp and ξ. The approximate solution for the extension xiis 8 Carlos A. Plata and Antonio Prados xi=`+ ξNkcδf(`)−vpγkc [3U00(`)+kc(N−1)]N(N+1) 6[Nkc+U00(`)] U00(`)[Nkc+U00(`)] +vpγkci(i−1)−2ξ[Nkc+U00(`)]δfi(`) 2U00(`)[Nkc+U00(`)] .(10) Here `6=L/N, it stems from the relation U0(`) = kc(L−N`).(11) We can see easily how we reobtain (6) taking the limit kc→∞in (10), as it should be. Although the solution is slightly different, it still breaks down when U00(`)vanishes, that is, when `→`b. Therefore, to the lowest order, again we have to seek a solution of (7), with the extensions given by (10), and substitute `c≃`btherein. This leads to the same critical velocities found for the infinite stiffness limit. 3.2 Location of the elastic reaction As depicted in Fig. 1, in an AFM experiment the distance between the moving platform and the fixed cantilever is the controlled quantity. Then, the model sketched in Fig. 3 is closer to the experimental setup: the left end corresponds to the fixed cantilever, with q0standing for ∆Zc, and the right end represents the moving platform. Thus, the free energy of this setup is given by A(q0,...,qN) = N ∑ i=1 Ui(qi−qi−1) + 1 2kcq2 0.(12) From the free energy (12), we derive the Langevin equations by making use of Eq. (1). The macroscopic equations (zero noise) read γ˙x1=−2U0 1(x1)+U0 2(x2)+kc L− N ∑ k=1 xk!,(13a) γ˙xi=−2U0 i(xi)+U0 i+1(xi+1)+U0 i−1(xi−1),1<i<N,(13b) γ˙xN=−U0 N(xN)+U0 N−1(xN−1)+vp.(13c) In the infinite stiffness limit, kc→∞, the harmonic contribution tends to a new Lagrange multiplier Fsuch that ∑xi=L. Therefrom, it is obtained that F=U0 1(x1) and the resulting system is exactly equal to that in Eq. (4). This is logical: if the spring is totally stiff and then the control over the length is perfect, the two models are identical.2 2It is worth emphasising that the two variants of the model, with the spring at either the fixed or moving end, have the same number of degrees of freedom. In Fig. 2, q0=0 and our degrees of Modelling the unfolding pathway of biomolecules 9 The system (13) can be solved in an analogous way, by means of a perturbative expansion in the asymmetry ξand the pulling velocity vp. The result is xi=`+ ξNkcδf(`)−vpγkc [3U00(`)+kc(N−1)]N(N+1) 6[Nkc+U00(`)] U00(`)[Nkc+U00(`)] + vpγkcii−1+2U00(`) kc−2ξ[Nkc+U00(`)]δfi(`) 2U00(`)[Nkc+U00(`)] .(14) Again, we can reobtain (6) taking the infinite stiffness limit in (14). Although the final solution for the extension is different from the previous one, when we look for the critical velocities and make the approximation `c≃`bwe get the same analytical results for them. Our main conclusion is that the existence of a set of critical velocities, setting apart regions where the first unit to unfold is different, is not an artificial effect of the limit kc→∞. Indeed, at the lowest order, all the versions of the studied model, independently of the value of the stiffness and the location of the spring, give the same critical velocities. This robustness is an appealing feature of the theory, and makes it reasonable to seek this phenomenology in real experiments. 4 Shape of the bistable potentials Different shapes for the double-well potentials have been considered in the literature, mainly simple Landau-like quartic potentials to understand the basic mechanisms underlying the observed behaviours [8, 9, 11] and more complex potentials when trying to obtain a more detailed, closer to quantitative, description of the experiments [9, 16–18]. For the sake of concreteness, we restrict ourselves to the proposal made by Berkovich et al. [16, 17]. Therein, the free energy of a module is represented by the sum of a Morse potential, which mimics the enthalpic minimum of the folded state, and a worm-like-chain (WLC) term [19], which accounts for the entropic contribution to the elasticity of the unfolded state. Specifically, the free energy is written as U(x) = U01−e−2bx−Rc Rc2−1+kBT 4PLc 1 1−x Lc −1−x Lc +2x2 L2 c!.(15) This shape has shown to be useful for some pulling experiments with actual proteins as titin I27 or ubiquitin [16, 17]. Therein, each parameter has a physical infreedom are qi,i=1,...,N, whereas in Fig. 3 we have the dynamical constraint qN=Land the degrees of freedom are qi,i=0,...,N−1. In the limit as kc→∞, we have both constraints, q0=0 and qN=L, in both models, making it obvious that they are identical. 16 Carlos A. Plata and Antonio Prados 18. L. L. Bonilla, A. Carpio, A. Prados, EPL 108, 28002 (2014) 19. J. F. Marko, E. D. Siggia, Macromolecules 28, 8759 (1995). 20. Q. Li, Z. N. Scholl, P. E. Marszalek, Angew. Chem. Int. Ed. 53, 13429 (2014) 21. I. Schwaiger, A. Kardinal, M. Schleicher, A. A. Noegel, M. Rief, Nat. Struct. Mol. Biol. 11, 81 (2003) 22. J. L. R. Arrondo, A. Alonso, Advanced Techniques in Biophysics (Springer, 2006) 23. G. Arad-Haase, S. G. 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