How the stability of a folded protein depends on interfacial water properties and residue-residue interactions
Valentino Bianco, Neus Pagès Gelabert, Ivan Coluzza, Giancarlo Franzese
aa r X i v : . [ c ond - m a t . s o f t ] A p r How the stability of a folded protein depends on interfacial water properties andresidue-residue interactions
Valentino Bianco , ∗ Neus Pag`es Gelabert , Ivan Coluzza , and Giancarlo Franzese , ∗ Universit¨at Wien, Sensengasse 8/10, 1090 Vienna, Austria Secci´o de F´ısica Estad´ıstica i Interdisciplin`aria–Departament de F´ısica de la Mat`eria Condensada,Facultat de F´ısica Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028, Barcelona, Spain and Institute of Nanoscience and Nanotechnology (IN2UB),Universitat de Barcelona, Av. Joan XXIII S/N, 08028 Barcelona, Spain (Dated: September 29, 2018)Proteins work only if folded in their native state, but changes in temperature T and pressure P induce their unfolding. Therefore for each protein there is a stability region (SR) in the T – P thermodynamic plane outside which the biomolecule is denaturated. It is known that the extensionand shape of the SR depend on i) the specific protein residue-residue interactions in the native stateof the amino acids sequence and ii) the water properties at the hydration interface. Here we analyzeby Monte Carlo simulations of different coarse-grained protein models in explicit water how changesin i) and ii) affect the SR. We show that the solvent properties ii) are essential to rationalize the SRshape at low T and high P and that our finding are robust with respect to parameter changes andwith respect to different protein models. These results can help in developing new strategies for thedesign of novel synthetic biopolymers. I. INTRODUCTION
The capability of the single components to indepen-dently organize in pattern and structures without an ex-ternal action fulfils a crucial role in the supramulecu-lar organization and assembling of the biological matter[1, 2]. To cite some examples, self-assembly is observedin bio–molecules [3], in DNA and chromosomes [4–7], inlipid membranes [8, 9], in the cytoskeleton [10], in cellsand tissues [11, 12], in virus and bacteria [13, 14], andin proteins [15, 16]. In particular, the protein foldingrepresents one of the most challenging and elusive bio-chemical processes where a chain of amino acids organizesitself into a unique native and folded structure [17, 18].The protein folding is a spontaneous process driven byintra-molecular (residue-residue) van der Walls interac-tions and hydrogen bonds which overcome the confor-mational entropy. It depends also on the presence ofco-factors as the chaperones [19] and, in particular, theproperties of the solvent, i.e. water [20], and the co-solutes [21] that regulate the pH level and the salt con-centration, for example.Although water has no influence on the primary struc-ture (the protein sequence), it affects the protein in allthe other level of organization [22–24]. Indeed, i) waterforms H-bonds with the polar/charged residues of theside chains, influencing the adoption of secondary struc-tures like alpha helices or beta sheets which expose themost hydrophilic residues to water; ii) the hydrophobiceffect drives the collapse of the protein core and sta-bilizes the tertiary protein structure; iii) water inducesthe aggregation of proteins since they usually presenthydrophobic regions on their surface (quaternary struc-ture). ∗ [email protected], [email protected] Experiments have clearly documented that proteinsmaintain their native structure in a limited range of tem-peratures T and pressures P [25–40] showing an elliptic-like stability region (SR) in the T – P plane, as accountedby a Hawley’s theory [41]. Outside its SR a protein un-folds, with a consequent loss of its tertiary structure andfunctionality.At high T the protein unfolding is due to the thermalfluctuations which disrupt the protein structure. Openprotein conformations increases the entropy S minimiz-ing the global Gibbs free energy G ≡ H − T S , where H is the total enthalpy. Upon cooling, if the nucle-ation of water is avoided, some proteins cold–denaturate[26, 28, 33, 35, 42–45]. Usually such a phenomenon isobserved below the melting line of water, although insome cases cold denaturation occurs above the 0 ◦ C, asin the case of the yeast frataxin [35]. Protein denatura-tion is observed, or predicted, also upon pressurization[25, 27, 34, 40, 46]. A possible explanation of the high- P unfolding is the loss of internal cavities, sometimespresents in the folded states of proteins [47]. Denatu-ration at negative P has been experimentally observed[48] and simulated recently [20, 48, 49]. Pressure de-naturation is usually observed for 100 MPa . P . P unless the tertiary structureis engineered with stronger covalent bonds [32]. Cold-and P -denaturation of proteins have been related to theequilibrium properties of the hydration water [20, 50–59]. However, the interpretations of the mechanism isstill largely debated [46, 47, 60–70].Here we investigate by Monte Carlo simulations ofdifferent coarse-grained protein models in explicit waterhow the SR is affected by changes in i) the specific pro-tein residue-residue interactions in the native state of theamino acids sequence and ii) the solvent properties at thehydration interface, focusing on water energy and densityfluctuations. In particular, after introducing the modeland the numerical method in Section II, we study in abroad range of T and P how the conformational spaceof proteins depends on the model’s parameters for thehydration water in Section III.A and how it depends onthe residue-residue interactions in Section III.B. Next,we discuss the possible relevance of our results in theframework of protein design in Section IV and, finally,we present our concluding remarks in Section V. II. MODELS AND METHODS
The extensive exploration with atomistic models ofprotein conformations in explicit solvent at different ther-modynamic conditions, including extreme low T and high P , is a very demanding analysis. To overcome thislimitation, we adopt a coarse-grain model for protein-water interaction based on A) the many-body water model[20, 59, 66, 71–79], combined with B) a lattice represen-tation of the protein.The many-body water model has been proven toreproduce–in at least qualitative way–the thermody-namic [71, 79] and dynamic [77] behavior of water, theproperties of water in confinement [72, 73, 76, 78] and atthe inorganic interfaces [66]. Its recent combination withthe lattice representation of the protein has given a novelinsight into the water-protein interplay [20, 59, 74, 75].As we will describe later, for the protein we considera model that, in its general formulation as polar pro-tein, follows the so-called “Go-models”, a common ap-proach in protein folding. In their seminal paper Goand Taketomi [80] employed non-transferable potentialstailored to the native structure. The interactions weredesigned to have a sharp minimum only at the nativeresidue-residue distance, guaranteeing that the energyminimum is reached only by the native structure. TheGo-proteins thus successfully fold, and have a smoothfree-energy landscape with a single global minimum inthe native structure [81]. Hence, Go-models are equiv-alent to having an infinite variety of pair interactionsamong the residues (alphabet A ), such that each aminoacid interacts selectively with a subset of residues definedby the distances in the native configuration. If the sizeof the alphabet is reduced, the construction of foldingproteins requires an optimization step of amino acid se-quence along the chain [62, 82, 83]; for this reason thesemethods are often referred as “protein design”. Compar-ing designed proteins with Go-proteins, Coluzza recentlyshown that, close to the folded state, Go and designedproteins behaves in a very similar manner [84]. Since weare interested in measuring the stability regions definedby the environmental condition at which the trial proteinis at least 90% folded, Go-models are an appropriate pro-tein representation, and, at this stage, we do not requireto perform the laborious work of protein design to ob-tain general results. We will discuss later the possibilityto extend our model to the case of a limited alphabet A of residues (20 amino acids). A. The bulk many-body water model.
We consider the coarse-grain many-body bulk waterat constant P , constant T and constant number N (b) ofwater molecules, while the total volume V (b) occupied bywater is a function of P and T . Because in the followingwe will consider the model with water at the hydrationprotein interface and (bulk) water away from the inter-face, for sake of clarity here we introduce the notationwith a superscript (b) for quantities that refer to thebulk.We replace the coordinates and orientations of the wa-ter molecules by a continuous density field and discretebonding variables, respectively. The density field is de-fined based on a partition of the available volume V (b) into a fixed number N = N (b) of cells, each with vol-ume v (b) ≡ V (b) /N (b) ≥ v , where v ≡ r is the waterexcluded volume with r ≡ . ρ (b) ≡ v /v (b) . As we will discuss later,the density is, instead, locally inhomogeneous when wa-ter molecules form HBs. Specifically, the density dependson the number of HBs, therefore ρ (b) only represents theaverage bulk density.The Hamiltonian of the bulk water is H (b) ≡ X ij U ( r ij ) − JN (b)HB − J σ N (b)coop . (1)The first term represents the isotropic part of the water-water interaction and accounts for the van der Waalsinteraction [85]. It is modeled with a Lennad-Jones po-tential X ij U ( r ij ) ≡ ǫ X ij "(cid:18) r r ij (cid:19) − (cid:18) r r ij (cid:19) (2)where ǫ ≡ . i and j at O–O distance r ij calculated as thedistance between the centers of the two cells i and j wherethe molecules belong. We assume a hard-core exclusion U ( r ) ≡ ∞ for r < r and a cutoff for r > r c ≡ r .The second term in Eq. (1) represents the directional(covalent) component of the HB, where N (b)HB ≡ X h ij i n i n j δ σ ij ,σ ji (3)is the number of bulk HBs and the sum runs over neigh-bor cells occupied by water molecules. Here we introducethe label n i = 1 if the cell i has a water density ρ (b) > . n i = 0 otherwise. In the homogeneous bulk this con-dition guarantees that two water molecules can form aHB only if their relative distance is r < / r ≡ . σ ij = 1 , . . . , q in Eq. (3) is the bondingindex of the water molecule in cell i with respect to theneighbor molecule in cell j and δ ab = 1 if a = b , or0 otherwise, is a Kronecker delta function. Each watermolecule has as many bonding variables as neighbor cells,but can form only up to four HBs. Therefore, if themolecule has more than four neighbors, e.g., in a cubiclattice partition of V (b) , an additional condition mustbe applied to limit to four the HBs participated by eachmolecule.The parameter q in the definition of σ ij is determinedby the entropy decrease associated to the formation ofeach HB. Each HB is unbroken if the hydrogen atomH is in a range of [ − ◦ , ◦ ] with respect to the O–Oaxes [87]. Hence, only 1 / , ◦ ] for the [ OOH angle is associated to a bondedstate. Therefore, in the zero-order approximation of con-sidering each HB independent, a molecules that has 4 − n HBs, with n = 1 , . . .
4, has an orientational entropy thatis S o n /k B ≡ n ln 6 above that of a fully bonded moleculewith S o0 /k B ≡
0, where k B is the Boltzmann constant.As a consequence, the choice q = 6 accounts correctly forthe entropy variation due to HB formation and breakinggiven the standard definition of HB.The third term in Eq. (1) is associated to the coop-erativity of the HBs due to the quantum many-body in-teractions [71, 88]. Indeed, the formation of a new HBaffects the electron probability distribution around themolecule favoring the formation of the following HB in alocal tetrahedral structure [89]. We assume that the en-ergy gain due to this effect is proportional to the numberof cooperative HBs in the system N coop ≡ X i n i X ( l,k ) i δ σ ik ,σ il , (4)where n i assures that we include this term only for liq-uid water. With this definition and with the choice J σ / ǫ ≪ J the term mimics a many-body interactionsamong the HBs participated by the same molecule. In-deed, the condition J σ / ǫ ≪ J guarantees that the in-teraction takes place only when the water molecule i isforming several HBs. The inner sum is over ( l, k ) i , indi-cating each of the six different pairs of the four indices σ ij of the molecule i .The formation of HBs leads to an open network ofmolecules, giving rise to a lower density state. We includethis effect into the model assuming that for each HB thevolume V (b) increases of v (b)HB /v = 0 .
5. This value isthe average volume increase between high-density icesVI and VIII and low-density (tetrahedral) ice Ih. As aconsequence, the average bulk density is ρ (b) ≡ N v V (b) + N (b)HB v (b)HB . (5)We assume that the HBs do not affect the distance r FIG. 1. Scheme of the water-protein coarse grain model. Theprotein is represented with red spheres. Each water moleculesis represented through its 4 bonding indexes σ , with differentcolours associated to the value 1 ...q assumed by σ . Direc-tional HB are represented with dotted lines joining two watermolecules. Cooperative bonds are represented with continu-ous lines connecting the σ indices inside a molecule. between first neighbour molecules, consistent with ex-periments [89]. Hence, the water-water distances r iscalculated only from V (b) .As discussed in Ref. [20] a good choice for the param-eters that accounts for the ions in a protein solution is ǫ = 5 . J/ ǫ = 0 . J σ / ǫ = 0 .
05 that givean average HB energy ∼
20 kJ/mol. In the following weconsider two protein models, a simpler one used to un-derstand the molecular mechanisms through which watercontributes to the unfolding, and a more detailed modelwhich includes the effect of polarization. For sake of sim-plicity, we present here the result for a system in twodimension. Preliminary results for the model in three di-mensions of both bulk water [79] and protein folding showresults that are qualitatively similar to those presentedhere.
B. Hydrophobic protein model.
The protein is modelled as a self-avoiding lattice poly-mer, embedded into the cell partition of the system. De-spite its simplicity, lattice protein models are still widelyused in the contest of protein folding [20, 51, 52, 58, 70,90–92] because of their versatility and the possibility todevelop coarse-grained theories and simulations for them.Each protein residue (polymer bead) occupies one cell.In the present study, we do not consider the presence ofcavities into the protein structure.To simplify the discussion in this first part of the work,we assume that (i) there is no residue-residue interaction,(ii) the residue-water interaction vanishes, unless other-wise specified and (iii) all the residues are hydrophobic.This implies that the protein has multiple ground states,all with the same maximum number n max of residue-residue contacts. As shown by Bianco and Franzese [20],the results hold also when the hypothesis (i), (ii) and (iii)are released, as we will discuss in the following.Our stating hypothesis is that the protein inter-face affects the water-water properties in the hydrationshell, here defined as the layer of first neighbour watermolecules in contact with the protein (Fig. 1). There aremany numerical and experimental evidences supportingthis hypothesis. In particular, it has been shown that thewater-water HBs in the protein hydration shell are morestable and more correlated with respect to the bulk HBs[93–98]. We account for this by replacing J of Eq. (1)with J Φ > J for the water-water HBs at the hydrophobic(Φ) interface. Another possibility, discussed later, wouldbe to consider that the cooperative interaction J σ, Φ atthe Φ-interface, directly related to the tetrahedral orderof the water molecules, is stronger with respect to thebulk. This case would be consistent with the assump-tion that water forms ice-like cages around Φ-residues[99]. Both choices, according to Muller discussion [100],would ensure the water enthalpy compensation duringthe cold-denaturation [59].At the Φ-interface, beside the stronger/stabler water-water HB, we consider also the larger density fluctua-tions with respect to the bulk, as observed in hydratedΦ-solutes [64, 95]. As a consequence, at ambient pressureΦ-hydration water is more compressible than bulk water.Although it is still matter of debate if, at ambient con-ditions, the average density of water at the Φ-interfaceis larger or smaller with respect to the average bulkwater density [101–105], there are evidences showingthat such density fluctuations reduce upon pressurization[64, 95, 106, 107]. We include this effect in the model byassuming that the volume change v (Φ)HB associated to theHB formation in the Φ hydration shell can be expandedas a series function of Pv (Φ)HB /v (Φ)HB , ≡ − k P − k P − k P + O ( P ) (6)where v (Φ)HB , is the value of the change when P = 0. Herethe coefficients k , k and k are such that ∂v (Φ)HB /∂P isalways negative. As first approximation, we study thelinear case, with k i = 0 ∀ i >
1. We discuss later how theprotein stability is affected by considering the quadraticterms in Eq.(6). Our initial choice implies that we canstudy the system only when
P < /k . As we will dis-cuss in the next section, this condition does not limit thevalidity of our results. The total volume V of the systemis, therefore, V ≡ N v + N (b)HB v (b)HB + N (Φ)HB v (Φ)HB . (7)where N (Φ)HB is the number of HBs in the Φ shell. C. Polar protein model
In order to account for the effect of the hydrophilicresidues on the water-water hydrogen bonding in thehydration shell, we consider also the case in which theprotein is modeled as a heteropolymer composed by hy-drophobic (Φ) and hydrophilic ( ζ ) residues. In this case isworth introducing residue-residues interactions that leadto a specific folded (native) state for the protein.We fix the native state by defining the interaction ma-trix A i,j ≡ ǫ rr if residues i and j are n.n. in the nativestate, 0 otherwise. To simplify our model we set all theresidues in contact with water in the native state as hy-drophilic, and all those buried into the protein core as hy-drophobic. The water interaction with Φ- and ζ -residuesis given by the parameters ǫ w , Φ and ǫ w ,ζ respectively,where we assume ǫ w , Φ < J and ǫ w ,ζ > J .The polar ζ residues interfere with the formation of HBof the surrounding molecules, disrupting the tetrahedralorder and distorting the HB network. Thus we assumethat each ζ residue has a preassigned bonding state q ( ζ ) =1 , ..., q , different and random for each ζ residue. In thisway, a water molecule i can form a HB with a ζ residue,located in the direction j , only if σ i,j = q ( ζ ) .In the polar potein model, the formation of water-water HBs in the hydration shell is described by theparameters i) J Φ and J σ, Φ (directional and cooperativecomponents of the HB) if both molecules hydrates twoΦ-residues; ii) J ζ and J σ,ζ if both molecules hydratestwo ζ -residues; iii) J Φ ,ζ ≡ ( J Φ + J ζ ) / J σ, Φ ,ζ ≡ ( J σ, Φ + J σ,ζ ) / ζ -residue,forming a Φ- ζ -interface. Accordingly, the volume asso-ciated to the formation of HB in the hydration shell is v (Φ)HB , v ( ζ )HB and v (Φ ,ζ )HB . Then, we assume that v (Φ)HB changeswith P following the Eq. (6). Due to the condition ǫ w ,ζ > J , we assume that the density fluctuations near a ζ -residue are comparable, or smaller, than those in bulkwater, therefore we set v ( ζ )HB = v ( b )HB . Finally, we define v (Φ ,ζ )HB ≡ ( v (Φ)HB + v ( ζ )HB ) / D. Simulations’ details
We study proteins of 30 residues with Monte Carlo sim-ulations in the isobaric-isothermal ensamble, i.e. withconstant P , constant T and constant number of parti-cles. Along the simulation we calculate the average num-ber of residue-residue contacts to estimate the proteincompactness, sampling ∼ independent protein con-formations for each thermodynamic state point. For thehydrophobic protein model, we assume that the proteinis folded if the average number of residue-residue con-tacts is n rr ≥ n max , while for the polar proteinmodel, having a unique folded state, we fix the thresholdat n rr ≥ n max .For sake of simplicity, we consider our model in two Temperature T [4 ε / k B ] -0,500,51 P r e ss u r e P [ ε / ν ] L i q u i d - G a s S p i n o d a l G l a ss T r a n s iti on
50% 30%Nativestate
FIG. 2. Stability region for a coarse-grained proteins madeof 30 hydrophobic residues. The dotted green line delimitsthe region within which the protein makes at least 30% ofits maximum number of contact points, i.e., n rr /n max ≥ . n rr /n max ≥ .
5, that by definition correspond to thenative state of the folded protein. The lower straight (black)dotted line represents the limit of stability (spinodal) of theliquid water with respect to the gas. The left-most (violet)solid line marks the limit below which water forms a glassstate. Adapted from Ref. [20]. dimensions. Although this geometry could appear as notrelevant for experimental cases, our preliminary resultsfor the three dimensional system show no qualitative dif-ference with the case presented here. We understandthis finding as a consequence of the peculiar property ofbulk water of having, on average, not more than fourneighbors. This coordination number is preserved if weconsider a square partition of a two dimensional system.Differences between the two dimensional and the threedimensional models could arise from the larger entropyin higher dimensions for the protein, however our prelim-inary results in 3D show that they can be accounted forby tuning the model parameters.
III. RESULTS AND DISCUSSIONA. Results for the hydrophobic protein model
Bianco and Franzese show [20] that the hydrophobicprotein model, with parameters k = v / ǫ (and k = k = 0), v (Φ)HB , /v = v (b)HB /v = 0 . J Φ / ǫ = 0 . J σ, Φ = J σ , has a SR that is elliptic in the T – P plane.This finding is consistent with the predictions of the Haw-ley theory [30, 41] accounting for the thermal, cold andpressure denaturation (Fig. 2).They find that at high T the large entropy associ-ated to open protein conformations keeps the protein un-folded. By isobaric decrease of T , the energy cost of an extended water-protein interface can no longer be bal-anced by the entropy gain of the unfolded protein, andthe protein folds to minimizes the number of hydratedΦ-residues, as expected.By further decreasing of T at constant P , the numberof water-water HBs increases both in bulk and at theprotein interface. At low-enough T , the larger stability,i.e., larger energy gain, of the HBs at the Φ-interfacedrives the cold denaturation of the protein.Upon isothermal increase of P , the enthalpy of the sys-tem increases for the increasing P V term. Therefore, amechanisms that reduces V would reduce the total en-thalpy. Here the mechanism is provided by the watercompressibility that is larger at the Φ-interface than inbulk. Therefore, the larger water density at the proteininterface drives the unfolding, which leads to a larger Φ-interface and enthalpy gain.Finally, when the system is under tension, i.e., at P <
0, the total enthalpy is minimized when V in Eq.(7)is maximized. However, the increase of average separa-tion between water molecules breaks the HBs. In partic-ular, bulk HBs break more than those at the Φ-interfacebecause the first are weaker than the latter. Hence, N (b)HB vanishes when N (Φ)HB >
0. As a consequence, the maxi-mization of V is achieved by maximizing N (Φ)HB , i.e., byexposing the maximum number of Φ-residues, leading tothe protein denaturation under tension.Once it is clear that the model can reproduce the pro-tein SR, allowing us to understand the driving mech-anism for the denaturation at different thermodynamicconditions, it is insightful to study how the SR dependson the model parameters. Therefore, in the following ofthis work we show our new calculations about the effectof varying one by one the model parameters.
1. Varying the water-water HB directional component J Φ at the Φ -interface. Changing the (covalent) strength J Φ of the interfacialHB has a drastic effects on the SR. As discussed above,having J Φ /J >
1, as in the reference case, drives thecold unfolding as a consequence of the larger gain of HBenergy near the Φ-interface. Instead, by setting J Φ /J < T then in the reference case, because there is a largerenergy gain in forming as many bulk HB as possible, i.e.,in reducing the number of those near Φ-residues. Hence,there is a larger free-energy gain in reducing the exposedΦ-interface with respect to the reference case.As a matter of fact, with our choice J Φ / ǫ = 0 .
20, wefind cold denaturation only for
P <
0. This is a conse-quence of the fact that the free energy has a term with N (Φ)HB multiplying ( − J Φ + P v HB − P v HB v / ǫ ), hence for P < N (Φ)HB increases, evenfor a vanishing J Φ . The negative slope of the cold denat-uration line at P < (a)
Temperature T [4 ε/ k B ] -0.500.51 P r e ss u r e P [ ε / v ] L i q u i d G a s S p i n o d a l G l a ss T r a n s iti on
70% 50% 30% (b)
Temperature T [4 ε/ k B ] -0.500.51 P r e ss u r e P [ ε / v ] L i q u i d G a s S p i n o d a l G l a ss T r a n s iti on % 30% FIG. 3. Effect on the SR of changing the water-water HBdirectional component J Φ at the Φ-interface. In both pan-els symbols with continuous lines delimit the regions with30% (green), 40% (turquoise), 50% (red) and 70% (blue) ofthe protein folded. Dashed lines (with the same color codeas for continuous lines) are for the reference system in Fig.2(Table I) with J Φ / ǫ = 0 .
55. All lines are guide for eyes.(a) For J Φ / ǫ = 0 .
20, smaller than the reference value, theSR expands to lower T and P and to higher T and P . (b)For J Φ / ǫ = 0 .
75, greater than the reference value, the SRshrinks. the larger | P | , the larger is the term proportional to N (Φ)HB in the free-energy balance.Reducing J Φ makes the folded protein more stable alsoat high T , because the entropy term overcomes the en-ergy term at T lower than in the reference case. A similarobservation holds also at high P , because a reduced J Φ implies a decrease in N (Φ)HB , hence a decrease in enthalpygain associated to the exposure of the Φ-interface.On the other hand, the larger | P | , the more negative isthe quadratic P -dependent coefficient that, as mentioned above, multiplies N (Φ)HB in the free energy, and the largeris the free-energy gain in exposing the Φ-interface at high T . Hence, the hot-denaturation curve in the P - T planehas a negative slope for P >
P <
0. As a consequence, the ellipsis describing the SR(Fig.3a for 50% curve) becomes more elongated than inthe reference case with a negatively-sloped major axisand an eccentricity that grows toward 1.On the contrary, for increasing J Φ the SR is lost, dueto the energetic gain associated to wetting the entire Φ-interface of the protein (Fig. 3b). The P -dependence ofthe contour lines is the same as discussed for the casewith J Φ /J <
1, hence they keep the shape but shrink.
2. Varying the water compressibility factor k at the Φ -interface. Decreasing the water compressibility factor k leads toa stretching of the SR along the P direction and a rota-tion of the ellipse axes in a such a way that the main axisincreases its negative slope in the P - T plane (Fig. 4a).On the other hand, increasing k results in a contractionof the SR along P with a rotation of the main axis towarda zero slope in the P - T plane (Fig. 4b).These effects can be understood observing that the freeenergy of the system has a term − k P N (Φ)HB . This termis associated to the fact that there is a larger water com-pressibility at the Φ-interface, reducing the total free en-ergy. Therefore, by decreasing k the destabilizing effectof the increased water-compressibility is reduced and theprotein gains stability in P at constant T , while the oppo-site effect is achieved by increasing k . The observationsabout the slope of the contour lines discussed in the pre-vious subsection apply also in this case explaining therotation of the ellipsis axes.
3. Varying the HB volume-increase v (Φ)HB , at the Φ -interfaceand P = 0 . A decrease of v (Φ)HB , /v , respect to the reference case,moves the SR at lower P , while an increase moves theSR at higher P (Fig. 5). This effect can be understoodobserving that the free energy of the system has a term P v (Φ)HB , N (Φ)HB that, at each P , implies a decreasing en-thalpy cost for decreasing v (Φ)HB , if N (Φ)HB is kept constant.Hence, this term favors the unfolding at high P when v (Φ)HB , is small, decreasing the stability of the native stateupon pressurization (Fig. 5a). The opposite occurs forincreasing v (Φ)HB , (Fig. 5b).We also find that the slope of the main ellipsis axischanges from positive, for small v (Φ)HB , , to negative, forlarge v (Φ)HB , . This is a consequence of the inversion of thecontribution of the free-energy term P v (Φ)HB , N (Φ)HB when P v (b)HB /v J/ ǫ J σ / ǫ v (Φ)HB , /v J Φ / ǫ J σ, Φ / ǫ k (4 ǫ ) /v k = k v ≡ r = 24 . and ǫ = 5 . (a) Temperature T [4 ε/ k B ] -0.500.51 P r e ss u r e P [ ε / v ] L i q u i d G a s S p i n o d a l G l a ss T r a n s iti on
50% 30% (b)
Temperature T [4 ε/ k B ] -0.500.51 P r e ss u r e P [ ε / v ] L i q u i d G a s S p i n o d a l G l a ss T r a n s iti on
50% 30%
FIG. 4. Effect on the SR of changing the water compressibilityfactor k at the Φ-interface. Symbols and lines are as in Fig.3and the reference system has k ǫ/v = 1. (a) For k ǫ/v =0 .
5, smaller than the reference value, the SR expands to awider range of P and the main ellipsis axis acquires a negativeslope in the P - T plane. (b) For k ǫ/v = 1 .
5, greater thanthe reference value, the SR contracts in P and the main ellipsisaxis becomes almost perpendicular to the P -axis. In bothpanels the effects of the change on the T -range of stabilityare minor. changes sign. Because a variation of v (Φ)HB , changes wherethe SR crosses the P = 0 axis, the stability contour-linechanges shape as a consequence, resulting in an effectiverotation of its elliptic main axis: the main axis is positivewhen the majority of the SR is at P <
4. Adding the quadratic P -dependence of v (Φ)HB at the Φ -interface. So far we have shown the SRs for the model with v (Φ)HB linearly-dependent on P . This truncation of Eq. (6) im-plies that the model for P < /k ≡ P L describes a sys-tem where water-water HBs at the Φ-interface decreasethe local density, as expected, while for larger P they dothe opposite. Thanks to our specific choice of parametersfor the reference system, our truncation does not affectsthe results because for P > P L the HB probability, bothin bulk and at the Φ-interface, is vanishing.However, to check how qualitatively robust are our re-sults against this truncation of Eq. (6), we consider alsothe case with the quadratic P -dependence of v (Φ)HB , i.e., v (Φ)HB /v (Φ)HB , ≡ − k P − k P , (8)where k > k /P .With this new approximation of Eq. (6) results P L ≡ (2 /x )( √ x − x ≡ k /k . Therefore, P L de-creases for increasing x .We fix k to the reference value, and vary k (Fig. 6).We find that for increasing k , the SR is progressivelycompressed on the high- P side, with minor effects on theSR T -range. Adding a cubic term in Eq. (6) affects theSR in a similar way (data not shown). The rational forthis behaviour lies in the enhanced enthalpic gain uponexposing the Φ-residue to the solvent since v (Φ)HB decreasesfaster upon approaching P L that, in turn, decreases forincreasing k .
5. Adding an attractive interaction ǫ w , Φ between water and Φ -residues. Here, we check how a non-zero water–hydrophobicresidue interaction, ǫ w , Φ >
0, would affect the SR of thehydrophobic homopolymer. Indeed, despite the commonmisunderstanding of “water-phobia” due to the oversim-plified terminology, it is well known that a hydrophobicinterface attracts water, but with an interaction that issmaller than a hydrophilic surface.We find that by setting ǫ w , Φ / ǫ = 0 .
05, smaller thanbulk water-water attraction, the SR is reduced in P andlightly shifted toward lower T (Fig. 7). In fact, an at-tractive water–Φ interaction enhances the propensity of (a) Temperature T [4 ε/ k B ] -0.500.51 P r e ss u r e P [ ε / v ] L i q u i d G a s S p i n o d a l G l a ss T r a n s iti on
50% 30% (b)
Temperature T [4 ε/ k B ] -0.500.51 P r e ss u r e P [ ε / v ] L i q u i d G a s S p i n o d a l G l a ss T r a n s iti on
50% 30%
FIG. 5. Effect on the SR of changing the HB volume-increase v (Φ)HB , at the Φ-interface and P = 0. Symbols and lines areas in Fig.3 and the reference system has v (Φ)HB , /v = 0 .
5. (a)For v (Φ)HB , /v = 0 .
1, smaller than the reference value, the SRmoves toward lower P and its main ellipsis axis rotates to-ward a positive slope in P - T plane. (b) For v (Φ)HB , /v = 1,greater than the reference value, the SR moves toward higher P rotates toward a negative slope in P - T plane. In both pan-els the effects of the change on the T -range of stability areminor. the polymer to expose the Φ residues to the solvent, re-sulting in a global reduction of the SR and destabilizingthe folded protein.
6. Enhancing the cooperative interaction J σ, Φ at the Φ -interface. Lastly, in the contest of the hydrophobic proteinmodel, we consider a different scenario. As discussedin the model description, the enthalpic gain upon cold
Temperature T [4 ε / k B ] -0.500.51 P r e ss u r e P [ ε / ν ] k =0.1k =1 L i q u i d - G a s S p i n o d a l G l a ss T r a n s iti on FIG. 6. Effect on the SR of adding the quadratic P -dependence of v (Φ)HB at the Φ-interface. Symbols and linesare as in Fig.3 and the reference system has k = v / ǫ and k = 0. For k (4 ǫ ) /v = 0 . P L v ≃ . k (4 ǫ ) /v = 0 . P L v ≃ .
90 and k (4 ǫ ) /v = 1 (orange circles and line)with P L v ≃ .
83, the SR shrinks at high P as P L decreases. Temperature T [4 ε / k B ] -0.500.51 P r e ss u r e P [ ε / ν ] L i q u i d - G a s S p i n o d a l G l a ss T r a n s iti on FIG. 7. Effect on the SR of adding an attractive interaction ǫ w , Φ between water and Φ-residues. Symbols and lines are asin Fig.3 and the reference system has ǫ w , Φ = 0. For ǫ w , Φ / ǫ =0 .
05, the SR moves toward lower T and shrinks in P . denaturation would be consistent also with the assump-tion J σ, Φ > J σ associated to a larger cooperativity of theHBs at the Φ-interface. Hence, to analyze this scenario,we compute the SR considering the directional compo-nent of the HB unaffected by the Φ-interface J Φ = J ,while assuming an enhanced HB cooperativity at the Φ-interface J σ, Φ > J σ . Note that the increase of J σ, Φ pro-motes the number of cooperative HBs at the Φ-interfaceonly once they are formed as isolated HBs ( J σ, Φ < J Φ ). Temperature T [4 ε /k B ] -0.500.51 P r e ss u r e P [ ε / v ] G l a ss T r a n s iti on L i q u i d G a s S p i n o d a l
50% 30%70%
FIG. 8. Effect on the SR of enhancing the cooperative inter-action J σ, Φ at the Φ-interface. Symbols and lines are as inFig.3. Here we adopted J σ, Φ / ǫ = 0 .
1, twice the value of J σ for bulk water molecules, while we fix J Φ / ǫ = J/ ǫ = 0 . k and v (Φ)HB , are as in Fig. 4a. Temperature T [4 ε / k B ] -0.500.51 P r e ss u r e P [ ε / v ] L i q u i d - G a s S p i n o d a l G l a ss T r a n s iti on FIG. 9. The SR for the polar protein model. We set theparameters as in Table II with all the other parameters asin Table I. Symbols with continuous lines delimit the regionswith 30% (green), 50% (red) and 80% (magenta) of the pro-tein folded. The other lines are as in Fig. 2. All lines areguides for eyes.
Our finding (Fig. 8) are consistent with a close SR, pre-senting cold- and pressure-denaturation.Although not discussed here, we expect that varyingthe parameters k and v (Φ)HB , , with the current choice of J σ, Φ > J σ and J Φ = J , would affect the SR similarly tothe cases discussed in previous subsections. (a) Temperature T [4 ε /k B ] -0.500.51 P r e ss u r e P [ ε / v ] L i q u i d G a s S p i n o d a l G l a ss T r a n s iti on
80% 50% (b)
Temperature T [4 ε /k B ] -0.500.51 P r e ss u r e P [ ε / v ] L i q u i d G a s S p i n o d a l G l a ss T r a n s iti on
80% 50%30%
FIG. 10. Effect on the SR of the polar protein of varyingthe residue–residue interaction ǫ rr . In both panels dashedlines are for the reference system in Fig. 9 (Table II) with ǫ rr / ǫ = 0 .
2, continuous (with the same color code as fordashed lines) are for the systems with a modified ǫ rr . (a) For ǫ rr / ǫ = 0 .
5, greater than the reference value, the SR expandsin P and T . (b) For ǫ rr / ǫ = 0 .
05, smaller than the referencevalue, the SR reduces in P and T . B. Results for the polar protein model.
Next we summarize the results for the polar proteinmodel. As shown in Ref. [20], also in this case the SRrecover a close elliptic–like SR in the T – P plane (Fig.9). In particular, despite we reduce the value of J Φ / ǫ with respect to the hydrophobic protein model in Ta-ble I, the additional residue-residue interaction ǫ rr andwater– ζ -residue interaction ǫ w ,ζ stabilize the folded stateto higher P and T , as can been seen by comparing Fig. 9with Fig. 2.0 Protein ǫ rr / ǫ ǫ w , Φ ǫ w ,ζ / ǫ J Φ / ǫ v ( ζ )HB J ζ / ǫ Polar 0.2 0 0.35 0.5 0 0.4TABLE II. Additional parameters for the reference systemsof the polar protein model (Fig. 9) with respect to those ofthe hydrophobic protein model in Table I. We also reduce thevalue of J Φ / ǫ with respect to Table I.
1. Varying the residue-residue interaction ǫ rr . To test how the residue-residue interaction ǫ rr is rel-evant for stabilizing the folded protein, we change itsvalue. We find that an increase of ǫ rr results in a broad-ening of the SR in T and P (Fig 10)a. We find the op-posite effect if we reduce ǫ rr (Fig 10b). These results areconsistent with our understanding that the native stateis stabilized by stronger residue-residue interactions.
2. Varying J Φ / ǫ and v (Φ)HB , at the Φ -interface. Next, we evaluate the effects of changing the water-water J Φ / ǫ interaction and the HB volume increase con-stant v (Φ)HB , at the Φ-interface for the polar protein model.We find that these changes affect the SR in a fashion sim-ilar to those discussed for the hydrophobic protein model(not shown). IV. PERSPECTIVE ON THE PROTEIN DESIGN
As we mentioned in the previous sections, the hydratedprotein models discussed here simplify the dependence ofthe stability against unfolding on the protein sequence.In fact, in the homopolymer protein model, the sequenceis reduced to a single amino acid, hence we have thealphabet A = 1, while in the polar protein model thealphabet size coincides, by construction, with the proteinlength l , A = l , because the interaction matrix has ( l − l ) / S [124]—and look for the protein sequenceswhich minimize the energy of the native structure. Thisscheme can be improved to account for the water prop-erties of the surrounding water, since the protein inter-face affects the water-water hydrogen bonding at leastin the first hydration shell. In this way, we aspect tofind sequences with patterns depending on the T and P conditions of the surrounding water. Our preliminary re-sults show that the protein sequences designed with ourexplicit-water model strongly depend on the thermody-namic conditions of the aqueous environment. V. CONCLUSIONS
In this work we have presented a protein–water modelto investigate the effect of the energy and density fluc-tuations at the hydrophobic interface (Φ) of the protein.In particular, we have considered two protein models. Inthe first we simplify the discussion assuming that the pro-tein is a hydrophobic homopolymer. In the second modelwe consider a more realistic case, assuming that the pro-tein has a unique native state with a hydrophilic ( ζ ) sur-face and a hydrophobic core and that the hydrophilicresidues polarize the surrounding water molecules. Inboth cases, we model the hydrophobic effects consideringthat the water–water hydrogen bond at the Φ-interfaceare stronger with respect to the bulk, and that the corre-sponding density fluctuations are reduced upon pressur-ization.Our model qualitatively reproduces the melting, thecold– and the pressure–denaturation experimentally ob-served in proteins. The stability region, i.e. the T – P region where the protein attains its native state, has anelliptic–like shape in the T – P plane, as predicted by thetheory [41].We discuss in detail how each interaction affects thestability region, showing that our findings are robust withrespect to model parameters changes. Aiming at sum-marize our findings, although the parameter variationsresults in a non trivial modification of the protein stabil-ity region, we observe that the strength of the interfacialwater-water HB compared to the bulk ones, mainly affectthe T –stability range of proteins, while the compressibil-ity of the hydrophobic hydration shell mainly regulatesthe P –stability range. The scenario remain substantially1unvaried by changing the protein model from the over-simplified hydrophobic homopolymer to the polar proteinmodel. Our findings put water’s density and energy fluc-tuations in a primary role to mantain the stable proteinstructure and pave the way for a water–dependent designof artificial proteins, with tunable stability. Acknowledgments
V.B. acknowledges hospitality at Universitat deBarcelona during his visits and the support from the Aus- trian Science Fund (FWF) project M 2150-N36. V.B.and I.C. acknowledge the support from the AustrianScience Fund (FWF) project P 26253-N27. G.F. ac-knowledges the support from the FIS2015-66879-C2-2-P(MINECO/FEDER) project. V.B., I.C. and G.F. ac-knowledge the support of the Erwin Schr¨odinger Inter-national Institute for Mathematics and Physics (ESI). [1] Jean-Marie Lehn. Toward Self-Organization and Com-plex Matter.
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