Reconfiguration dynamics in folded and intrinsically disordered protein with internal friction: Effect of solvent quality and denaturant

https://doi.org/10.1016/j.physa.2015.12.147Get rights and content

Highlights

  • We calculate reconfiguration time for a chain with internal friction.

  • The effect of solvent quality has been taken care of by a parameter ν.

  • The reconfiguration time scales as τN2ν+1, where N is the chain length.

  • Our calculations are in excellent agreement with the recent experiments on proteins.

Abstract

We consider a flexible chain with internal friction in a harmonic confinement and extend it to include the effects of solvent quality at the mean field level by introducing a Flory type exponent ν. The strength of the harmonic confinement (kc) accounts for the denaturant concentration and connects to the internal friction of the chain (ξint) through an ansatz. Our calculated reconfiguration times falling in the range of 5–50 ns are found out to be within 10%–15% of the experimentally measured reconfiguration times of the folded cold shock protein and the intrinsically disordered protein prothymosin α. In addition, our calculations show that the reconfiguration time scales with the chain length N as Nα, where α depends weakly on the internal friction but has rather stronger dependence on the solvent quality. In the absence of any internal friction, α=2ν+1 and it goes down in the presence of internal friction, but chain reconfiguration slows down in general. On the contrary, in a poorer solvent chain reconfiguration and looping become faster even though the internal friction is higher in the collapsed state.

Introduction

Among the polymer rheologists the notion of internal friction associated with a single polymer chain is more than twenty five years old  [1], [2], [3], [4]. Surprisingly it is only very recently that the topic has gained attention in the chemical and biophysics community. Although some earlier works in chemical physics community did point out the importance of internal friction in single chain dynamics  [5], [6] but did not receive the attention it should have. However recent experiments on looping and folding dynamics  [7], [8], [9], [10], [11], [12] in proteins have indicated the non-negligible role of internal friction. As for example, a plot of reconfiguration time of cold shock protein against solvent viscosity results in a non-zero intercept, a signature for the existence of solvent independent friction termed as “internal friction”  [11]. Similarly the protein folding rates have shown to deviate from a Kramer’s type behavior with a fractional viscosity dependence and generally attributed to the presence of a friction within the protein itself  [13]. In this context it is worth mentioning that loop formation between any two parts of a bio-polymer  [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] is supposedly the primary step of protein folding, DNA cyclization and since internal friction affects looping it is obvious that measurements of folding rates in proteins would predict the importance of solvent independent internal friction as well  [26]. Subsequently not only these recent experiments  [11], [27], [12] on polypeptides and proteins showed internal friction to play a pivotal role in the dynamics but also motivated theoretical polymer scientists to come up with statistical mechanical models for single polymer chain with the inclusion of internal friction  [28], [29], [30], [31], [32] and apply these models to investigate the loop formation dynamics. Other than model build up there have been attempts to elucidate the origin of internal friction in proteins based on pure computer simulation studies  [33], [34], [35], [36], [37], [38], [39]. However a careful literature survey would reveal that it was de Gennes  [40], who introduced the concept of internal viscosity at the single chain level. Rabin and Öttinger proposed an expression for the relaxation time, τrel associated with internal viscosity following an idea of de Gennes  [40], which is τrel=R3/kBT(ηs+ηi) where, R=aNν and a,N are the monomer size and chain length respectively, ν is the Flory exponent  [41], [42], [43]. Therefore in the limit solvent viscosity ηs0, it has a non-zero intercept proportional to the internal viscosity ηi. This is what exactly seen in recent experiments where the plot of reconfiguration time vs solvent viscosity has a finite intercept equal to the time scale for the internal friction  [11].

To the best of our knowledge so far all the theoretical attempts on loop formation in a single chain with internal friction have been restricted to θ solvent, with ν=1/2, when the chain behaves ideally. But the experimental conditions remain close to a good solvent rather than a θ solvent. There have been few theoretical studies to elucidate the effect of solvent quality on loop formation in single polymer chain  [44], [45], [46] and unfortunately there exists almost no theoretical study to analyze the combined effect of solvent quality and internal friction on the ring closure dynamics in polymer chains other than the very recent simulation by Yu and Luo  [47]. Apart from the solvent quality, denaturant concentration does play an important role in ring closure dynamics  [32], [48] of proteins as it profoundly affects the compactness of the protein and routinely used in experiments. Nuclear magnetic resonance and laser photolysis methods also have confirmed the effects of denaturants by showing the rate of intrachain contact formation in unfolded state of carbon monoxide-liganded cytochrome c (cyt-CO) to increase with the increase in denaturant concentration  [12].

Recently Samanta and Chakrabarti  [32] used a compacted Rouse chain model with internal friction to infer the role of denaturant on ring closure dynamics. But the model works only in the θ solvent condition where a phantom Rouse chain description holds. Experiments have been performed with proteins away from the θ conditions where excluded volume interactions along with the internal friction play important roles. To take care of excluded volume interactions one has to go beyond phantom chain description but then the many body nature of the problem does not allow an analytical solution. One possibility would be to perform computer simulation as is recently done by Yu and Luo  [47]. Other possibility would be to work with a polymer chain where the excluded volume interactions are taken care of at the mean field level. We take the second route and propose here a very general analytical model that takes care of solvent quality as well as denaturant in addition to internal friction. We call it “Solvent Dependent Compacted Rouse with Internal Friction (SDCRIF)”. To take care of solvent effect we closely follow the work of Panja and Barkema  [49] where an approximate analytical expression for the end to end vector correlation function for a flexible chain in an arbitrary solvent was proposed based on a series of computer simulations. The expression carries a parameter ν similar in the spirit of Flory exponent  [41], [42], [43], [50], [51] which takes care of solvent quality. A value of ν=1/2 corresponds to a θ solvent and in that case the correlation function is exact and reduces to the text book expression for the ideal chain  [52], [53], on the other hand ν=3/5 (0.588 more precisely)  [49] corresponds to a self avoiding flexible chain (good solvent) as is the case with real polymers. Importantly the same expression can be used for a range of values of ν corresponding to different solvent qualities. Very recently such a mean field Flory exponent based model has been used to describe ring polymer dynamics  [54].

Next is the inclusion of internal friction which is done similarly as in case of a phantom polymer chain  [55]. Further to take care of the denaturant which controls the compactness of the protein a confining harmonic potential with force constant kc is used as is done in a very recent study by the authors  [32] and also in the context of diffusing polymers in microconfinements  [56], or in case of bubble formation in double stranded DNA  [57]. So the novelty of SDCRIF remains in its applicability for a range of solvent quality, denaturant concentration and in addition it also takes care of internal friction (ξint) when required. SDCRIF reduces to Rouse model in the limit ν=1/2,kc=0,ξint=0 and to “Compacted Rouse with Internal Friction (CRIF)”  [31] in the limit kc0,ξint0,ν=1/2. Thus SDCRIF can be viewed as the most general model and other models are special cases of SDCRIF. The looping dynamics is studied within Wilemski Fixman (WF) framework   [14], [58] with SDCRIF, assuming the polymer chain to be Gaussian. It is worth mentioning that WF formalism has extensively been used to calculate the average ring closure or looping time in presence of hydrodynamic interactions by Chakrabarti  [59] and to elucidate the effect of viscoelastic solvent  [60], [61] and even applied to ring closing of a semiflexible chain  [62].

The arrangement of the paper is as follows. The model is introduced in Section  2. Section  3 deals with the method used to calculate the reconfiguration and ring closure time. Results and discussions are presented in Sections  4 Results, 5 Conclusions concludes the paper.

Section snippets

Solvent quality dependent compacted Rouse with Internal friction (SDCRIF)

In the Rouse model, the polymer chain is treated as a phantom chain  [52], [53], where the hydrodynamics interactions and the excluded volume effects are not present. If Rn(t) denotes the position of the nth monomer at time t of such a chain, made of (N+1) monomers, n varies from 0 to N. The equation describing the dynamics of the chain is the following ξRn(t)t=k2Rn(t)n2+f(n,t), where ξ denotes the friction coefficient and k=3kBTb2 is the spring constant where b is the Kuhn length. f(n,t)

Reconfiguration time

The time correlation function of the end-to-end vector is calculated from the correlation of normal modes as follows ϕN0(t)=RN0(0)RN0(t)IV=16p=13kBTkp,IVexp(t/τp,IV).

Therefore, ϕN0(0)=RN02eq,IV=16p=13kBTkp,IV. The exact expression for RN02eq,IV is not analytically trackable. However in θ solvent (ν=1/2), it has an analytical expression  [32]. RN02eq,ν=1/2,V=2b3kBTkctanh[Nbkc23kBT].

Here V stands for CRIF.

Fortunately for SDRIF, RN02eq,II has an analytical expression too RN02eq,II

Equilibrium end to end distribution

The equilibrium distribution of the vector connecting end-to-end monomers of a Gaussian chain is given by P(RN0)=(32πRN02)3/2exp[3RN022RN02], where, RN02 denotes the average equilibrium end to end distance of the polymer. This expression holds for SDCRIF as well since the model is considered to be Gaussian in our description. In Fig. 2 the equilibrium distribution of the end-to-end distance is shown for the polymer at different solvent quality in the absence of the confining potential.

Conclusions

Motivated by recent experiments  [11] on cold shock proteins and intrinsically disordered proteins (IDPs) we have analyzed the effects of denaturant and the solvent quality on the reconfiguration and looping dynamics of a chain with internal friction by using an extended Rouse chain model with internal friction. The model termed as “Solvent Dependent Compacted Rouse Chain (SDCRIF)” takes care of solvent quality through a Flory type exponent ν and the effects of denaturant concentration are

Acknowledgments

Authors thank IRCC IIT Bombay (Project Code: 12IRCCSG046), DST (Project No. SB/SI/PC-55/2013) and CSIR (Project No. 01(2781)/14/EMR-II) for funding generously.

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