Protein-Involving nearby Velocity Field

22 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. Figure 3. Protein crystallization by LBS thin film nanotemplate: LBS nanotemplate, used to modify classical hanging drop method [43], trigger the specific aggregates’ forma- tion in the protein drop solution and thereby helps to overcome the nucleation free energy barrier, see Fig. 13. e.g. curvature terms it may contain or gives rise to readily nonequilibrium effects at the boundary, especially the ones coming from elastic interactions [39] between the boundary proteins and its nearby electrolyte surroundings also, within the boundary when some addi- tional strain between the vacancies may arise, or when spherulites composed of fibrils tend to emerge, even with a tendency of lateral growth violating here somehow the continuous- growth mode invoked by Cahn which suits also our type of reasoning [41]. Thus, the overall nonequilibrium boundary condition can be proposed as follows [39, 42] c ne R = c o h 1 + Γ D × K 1 R + δ T 2 × K 2 R + Σ 3 i =1 α i x i R − β k × dR dt i , 6 where δ T Tolman length, K 2 r = 1R 2 Gaussian curvature, α i the elastic coefficients and x i R = R i − R i o R i o R o is the initially taken object’s radius, with the latter representing three main crystal-surface nucleation mechanisms [39]. β k stands for the kinetic coefficient, and dR dt points to the readily nonequilibrium character of the boundary condition [42], expected to occur for spherulites, especially those of lysozyme type. Because of stating Eq. 6 in such a general form one is then able to write down a more general kinetic equation [39] d dt R = σ R ne × v mi , 7 where σ R ne = c ne R C − c ne R 8 under a necessary restriction that all nonequilibrium mechanisms prescribed at the bound- ary, cf. Eq. 6, do not need to operate at the same time together. They can rather be switched onoff whenever physically legitimate [44, 40, 41, 45].

2.1.3. Protein-Involving nearby Velocity Field

In the deterministic description one may notice the Frenkel-type velocity of the macroion incorporated in the resulting equation, see v mi in Eq. 7. In order to achieve a more real- Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 23 istic description of the process one is motivated to offer a correlational proposal for v r, t. Although it can separate into its spatial r and temporal t parts, in the very vicinity of the crystal, i.e. mostly in the diffusive part of its Stern-type double layer the spatial [42] cor- relations can be postponed since they are hard to detect in such a narrow depletion region, especially when the ballistic motion of the proteins is foreseen. In turn, the temporal corre- lations in v r, t cannot be ruled out. They arise because the viscosity in the diffusive part of the Stern double layer decreases in time when the crystal grows. Since the crystal becomes bigger the double layer expands too, and because the external-concentration conditions are assumed unchanged, a protein-influenced viscosity change near the crystal surface results in some viscosity decrease in time. This can be described as a depletion-zone effect around the growing object. It is due to some diffusivity, D t, effective increase in time which comes from the time correlations in v r, t to be inferred from the Green-Kubo formula, i.e. when the fluctuation-dissipation theorem can be applied [33]. Thus, after postponement of the spatial correlations, the correlational proposal may be a stationary Gaussian but correlated, though with a zero average velocity part vt = 0, K | t − s | = vtvs for t 6= s, 9 where the time correlations K s ′ , with the characteristic correlation time τ cor , given by means of an average · · · , are related to Dt by D t = t Z K s ′ ds ′ for s ′ ≥ 0. 10 It is found out that the most interesting correlational proposal to be offered here appears to be the following [46] D t ≃ τ cor 1 − γ × t τ cor 1 − γ for t ≫ t o , 11 where γ ∈ 0, 1 is a characteristic fractional exponent, and t o is taken as an initial time. It can be anticipated that [42], if γ → 0 is obeyed, the rounded protein-crystal formation points to an overall ballistic, hydrodynamically stable growing asymptotic mode, since D t ∝ t eventually applies. It greatly contributes to the total hydrodynamic stability of the forma- tion, especially in a sufficiently mature growing stage see, Fig. 4, since for t t o the late-time solutions of the stochastic d dt R = σ R ne × vt, 12 cf. relations 7 and 9, R ≡ Rt, are given by R ∼ t, pointing to on-average-constant speed of the formation dR dt → const., which is true because σ R ne typically suppresses to a constant value, being an inverse of the supersaturation, when t t o , and v t → v mi = const . on average as well. There is another signature of the ballistic character of the process, herein at the meso- scopic level. At a “macroscopic level” of description one may see the relation given by Eq. 4 as some signature of a ballistic character of the process, whereas its microscopic 24 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. counterpart corresponds to Eq. 11 correlation strength with γ → 0. The real challenge that can happen here is, however, that Eq. 12 is fully equivalent to a Smoluchowski-type equa- tion [36] in R–configurational space that can be solved with suitable viz reflecting boundary conditions characteristic of a two-state dynamic process with a surmountable Kramers-type energetic barrier [47]. This barrier, given by Φ , can be determined exactly, and both its height and shape can be controlled by the fluctuations [45] of the radius of the SPC-A given by the diffusion function involved in the Smoluchowski-type dynamics, D R, t = D[Rt], defined still under t t o as D [Rt] = ∞ R R 2 ρ R, tdR 2t , 13 whereby the above relation is the Einstein-type relation in the one-dimensional phase space R.

2.1.4. Smoluchowski SPC-A Scenario in