38 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
Before analyzing the transport process from the point of view of numerical simula- tions, it is convenient to stress that our description can be complemented with that already
presented in Secs. 2.1.1.-2.1.9.. Other generalizations are still possible when there in a ve- locity gradient driving the whole system out from equilibrium. Such a case will be studied
in Sec. 3.5..
2.1.11. Diffusing Probe-Particle Spectroscopy Applied to Reveal Nonlinear Vis-
coelastic Properties of Biomatter - a Computer Experiment Diffusion-type phenomena such as the Random Walk RW of a probe particle can be suc-
cessfully simulated using Monte Carlo MC methods. In order to test a RW along a crystal terrace made of H hydrophobic and P polar sites on a square lattice, a special prepa-
ration of a h = 20 columnar deposit h-height has first been undertaken [84]. Next, the
HP properties of the h = 20 deposit have been mapped onto a plane, this way constituting
a planar terrace of the crystal. Each point of the terrace either H or P property has been assigned by applying a majority rule to each site labeled by i: a site
i =H if number of H at sitecolumn i is greater than a number of P at the same sitecolumn. If this numbers are
equal, a random sampling decided for the proper assignment to i. Afterwards, the terrace’s surface has been restructured in order to get a percolation track, built up of P sites a certain
necessary replenishment of P sites, becomes effective, since the surface is always exposed towards water.
Figure 7. Crystal’s terrace used in computer experiment with diffusing probe particle; gray: H-hydrophobic site, white: P-hydrophilic site
A RW of neutral testing walker has been carried out along so prepared N × N terrace,
with maximum N = 1024, see Fig. 7. First, the one-half diffusion exponent, either with
elasticreflecting - case A confined space or with periodic - case B unconfined space boundary conditions b.cs., or both of them, has been approached and reflected properly
by a scaling law of
N ∼ t
ν
d
, 54
where t-time, and ν
d
-RW exponent, in log − log scale. The reflecting b.cs. indicate an
existence of external force which confines the particles’ motion. The periodic b.cs, in turn,
Can Modern Statistical Mechanics Unravel Some Practical Problems . . .
39 remove this confinement and make that only internal forces are included into the system.
Then, a small few lattice constants lasting drift has been superimposed on the RW, favor- ing a bit longer glides over sub-chains of P sites, whereas glides over H sites were always
forbidden.
Figure 8. Super- left versus sub-diffusion right RW characteristics derived from a com- puter simulation in which a testing hydrophilic-hydrophobic particle explores by a RW,
equipped with the hydrophobic type interaction, a crystal’s terrace which is a 2 D matrix composed of amphiphilic residues.
Figure 9. Left plot: RW exponents ν
d
for flight lengths equal 2 ÷ 6 for p =
P P
+H
= 0.7, and when performed for the case without drift p
= 0.5; right plot: RW exponent ν
d
for p
= 0.7 and particle’s flight equal 4 lattice constants. The simulations were executed on N
× N terrace, with maximum N = 100. For slightly drifted RW, the
ν
d
has been obtained as ∼ 0.7, see Fig. 8a. Finally, a HP
changing-property, let us say, “flicker” walker has been allowed to move randomly along the HP matrix 51
.7 P sites on the matrix, additionally residing 4 times units when H −H interaction occurred penalty=4, 2 times units when H
− P or P − H interaction took place penalty=2, and finally 0 time units penalty=0 when P
− P interaction resulted. Then, ν
d
exponent has been carefully derived to be ca. 0 .4, slightly depending on the b.cs. applied,
cf. Fig. 8b, and suffering a bit from a finite-size effect. This way, a passage between super- and subdiffusion along the terrace can be realized, and some reasonable control over it can
also be assured [8]. For modified RW, in which besides HP changing-property we add drift obtained by the privilege of the movement in the right-hand side target direction, setting up
probability of the movement twice so big than in left direction and adding a glide over the
40 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
path of 4 sites length of the flight is equal to 4, the ν
d
has been carefully obtained as ∼ 1,
see Fig. 9. Such realizations have already been tested experimentally by means of diffusion wave
spectroscopy [64, 83]. They confirmed a fully viscoelastic behavior of the tested matrix characteristic of the exponents such as those coming from our simulations, see Fig. 8. But
above all they confirm that the HP “flicker” walker quite typically performs a superdiffusive RW along a crystal’s terrace of the same physical nature. The RW exponent of such
realization equals one, as in case of analytic studies [14].
2.2. Theory and Experiment for Confined SPC-A and Protein
