MNET in an Effectively 2D Computer Experiment on Model Biopolymer Ag-

Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 51

2.3.2. MNET in an Effectively 2D Computer Experiment on Model Biopolymer Ag-

gregation As we mentioned in Sec. 2.3.1., formation of micelles and growth by protein aggregation can be described within the MNET framework. In the aqueous protein solution, which we may assume as a viscoelastic environment, two concurrent effects confront: diffusion, caused by the random thermal motions of the proteins in solution; and drift, resulting from hydrophobic-hydrophilic viz structural properties [13] of interactions between aminoacids cf., Eq. 14. Both of them are, however, permanently coupled to aggregation events occurring in the inhomogeneous aqueous solution of interest. Thus, the interactions are leading to the biopolymer-chain folding and some binding acts [106]. Relying on the observations on a kinetic-thermodynamic optimality of the micellar growth pace in two-dimensional 2D; d E = 2 space, with an interaction-caused excluded- area effect [107], also very characteristic of emerging 2D micellar assemblages, we have conducted in a comparative way the effectively 2D 4 of micelles’ formation. The basic aim of the experiment performed was to unravel in which kinetic-thermodynamic conditions the structures, like rounded 2D polycrystals, or their purely amorphous counterparts [57] appear, and how to control their features, as well as - as mentioned above - to compare the results obtained within the analytic MNET model. By virtue of lack of any external perturbation applied to the system, the 2D micelles’ formation qualifies as a good example of IFS. It is mainly due to the inhomogeneity of the by-definition viscous system that we examine [105]. This inhomogeneity propagates over all computer-simulation time read, thus qualifying the entire system under study as microrheological [10]. Therefore, it is also worth noting here that internal forcestensions are exclusively a result of typically symmetry-breaking [13] interactions amongst H andor P residues. The HP particles, because of their self-conflicting viz hydrophobic properties, resting on their hate-love relationship to water, being even implicitly stated in classical Dill’s model [108], are attaching to or detaching from each other during its random motions, showing clear signs of frictional inhomogeneities even more. Our approach is based on the possibly simplest although effective 2D lattice model called the tube hydrophobic-polar T HP model [109], utilizing a MC algorithm for com- puter simulations of protein folding. T HP model, contrary to standard HP model [98], emphasizes proper folding conditions, which favors a long-range fibrillar order within each model HP chain. We extended the offered model as being applicable to the aggrega- tion of HP linear chains of self-avoiding character [107], and possibly, having it applied to clusters made of such chains [105, 110], see Fig. 15. From this computer experiment, one is also able to get some typical cluster - clus- ter aggregation CCA characteristics [111], e.g., an average number of HP clusters in- volved in the CCA; an average radius, characterizing a CCA under examination, and an average area covered by CCA, all over examined in the course of computer-simulation time, in MC steps [105] and Figs. 5-7 within. Fortunately, the above are also princi- pal characteristics of some percolation sol-gel system such as the one prone to gelation [112, 38, 111, 113]. To the first approximation, all characteristics mentioned are fairly ac- 4 Effectively 2D means that every HP chain is placed in the 2D Euclidean space but HP particles are allowed to escape when moving into third space dimension for a short time period. 52 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. Figure 15. An example of simulation result of the rounded 2 D polycrystals formation during the folding and aggregation processes in model biopolymer system. HP chains H - red dots, P - blue dots assemble into clusters. Left picture represents an initial nearly homogeneous state with A R = 1, cf. Eq. 66 and right picture the inhomogeneous state after 200,000,000 Monte Carlo steps MCs; 0 AR 1. cessible by Smoluchowski-type dynamics [114, 36], in our approach applied in terms of MNET [114], working preferentially in the space of the cluster sizes R. Their properties are able to tell us with which formations of the model biopolymer clusters polycrystals andor disordered aggregatesmicelles we are actually dealing [38]. The preliminary results of the computer experiment performed so far [105, 107] in- dicate: i clearly emerging possibility of judging when the system is in a sol state, and when it makes an access to its gelling counterpart; ii straightforward accessibility of the sol phase by its more or less crystalline propensity, and the gel phase by its rather fairly amorphous provenience, the latter being also more exposed outwards towards water when compared to the former, being always more self-focused viz close-packed around its ag- gregation center see, Fig. 2 in [105]; iii quite large discrepancy between analytical and computer model, mainly due to still pending adjustment of computer- and “analytic” time domains, as well as owing to putting too much emphasis on favoring detachment events over their attachment counterparts just in the computer model - this way a step towards physically realistic design of the process has been proposed [115].

3. Second Example: The External-Friction System EFS

Explained in Terms of Dissipative Structure Called Model Articular Cartilage Necessary Prelude While embarking on reading this part, especially Secs. 3.A-C, one has to be aware that the material presented in Sec. 3.A-C has been thought of to be an extensive survey of experi- mental facts and evidences on the AC, clearly referring to its realistic, biomedical charater, whereas Sec. 3.4. is allowed for abstracting a little bit from such a cumbersome albeit beautiful reality of the system under consideration. It is in turn attempting to perform a StatMech modeling of the complex AC decribed, which is an extremely subtle CA system, abstracting in a certain way from some redundancy of almost all “necessary” details re- sponsible for the system’s rheological behavior. By the way this, let us say, redundancy