Can Modern Statistical Mechanics Unravel Some Practical Problems . . .
45 where t - as above in Eq. 54; note that
χ
cc
∝ G
′′−1
holds finally. It is interesting to note that when
ν
d
≈
3 8
, one gets the most common temporal behavior of G
′′
, namely G
′′
∼ t
−
3 4
, that represents linearly elastic harmonic response of the medium, e.g. that of the cytoskeleton [8]. This temporal property translated to frequency domain gives G
′′
∼ ω
i
3 4
, the most characteristic case of material’s linear viscoelastic response; bear in mind that
ν
d
=
3 8
is sometimes called a de Gennes’ slightly-constrained diffusional exponent, cf. [93], or refer to original literature [94]. Let us point out, that the small-scale simulations presented
show, that a non-Kossel amphiphilic crystal emerges when ν
d 3
8
, distinctly, attaining exactly the limit of
ν
d
= 1, just a ballistic-growth exponent [14]. From the above it can be concluded that, for example, an amphiphilic aggregate grows
as a crystal in a terraces-involving way, when on average the creep compliance, χ
cc
, of the representative terrace responds in a quadratic fashion, or “nonlinearly”, in time. In other
words, the complex modulus of the ’average’ terrace should behave with the interaction frequency,
ω
i
, as G
′′
∼ ω
i 2
, cf. Fig. 6. Otherwise, when G
′′
∼ ω
i ε
, with ε
6= 2 holds, the terrace’s material is either too soft
ε 2 or too stiff
ε 2 , thus, not responding
properly in a crystal-like manner, which is ultimately to conclude, that an unorderly ag- gregate is presumably being formed, cf. the b.cs. of the analytic model [14], and refer
to b.cs. given by Eq. 6. Note that the viscoelastic limit of growingyielding protein or, in general, amphiphilic crystals is still immersed within the soft-response range of
ε 2.
It is, however, worth noting that only large-scale computer simulation [95] can verify thor- oughly the values of the exponents derived. If they would have been successful, a combined
analytic-simulational viz hybrid model could have become a powerful tool.
2.2.4. Nanocrystallography: New Theoretical Approach
A modern experimental technique, such as protein nanocrystallography, has recently been nominated to be a unique nanotechnology-based method of stable protein crystals formation
and their characterization down to atomic scale. In particular, N-G transformation of so far unsolved protein can be simulated by protein nanostructured template [16, 15]. In case of
lysozyme, a crystal growth appears to transfer directly from the nanostructured film into the drop, being the first step in crystallization process [15], see Fig. 3.
Figure 13. Energy-barrier overcoming in protein crystallization process. In a letter [43] it was demonstrated for the first time that a new experimental nanocrys-
tallographic approach allowed someone to accomplish a visible increase of the hen egg white lysozyme HEWL N-G rate in comparison with such a classical vapor diffusion
method as hanging drop [96]. The approach relied on a modification of the classical vapor
46 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
diffusion method with the aim of crystal growth acceleration. A HEWL Langmuir-Blodgett LB thin film, prepared by Langmuir-Schaefer LS technique variation, was used as the
template for the stimulation and rate increasing of lysozyme crystal growth [97]. Monolay- ers of lysozyme were formed in a Langmuir Teflon trough by spreading 500 ml phosphate
buffer pH 6
.5 solution with a HEWL concentration 4 mgml; 10µM NaOH solution pH 11 was used as a subphase. The subphase temperature was 22
◦
C. The formed film was compressed with a barrier speed of about 0
.1mms up to surface-pressure of 18mN m and deposited by LS parallel shift technique onto the siliconazed cover glass slide. Obtained
nanofilm was characterized by circular dichroism, atomic force microscopy and nanogravi- metric methods [97] and utilized as a template for crystal growth in a common crystalliza-
tion apparatus, placed in a contact with a protein solution drop, see [16, 15, 43, 96].
The applied computer model points fully to an interface-controlled towards thin films process - we propose our computer-model based confirmation of the experimental fact that
the crystal growth based on 2AUB PDB ID is by a factor
4 3
faster when compared to the same type of N-G process under the same basic thermodynamic-kinetic conditions
[16, 15, 43, 18] but performed on a 193L PDB ID lysozyme. In what follows we will develop our type of argumentation that slightly modifies the before applied algorithm of
simulation [18] in order to achieve a result being in excellent accord with the performed experiment. Our main concern is that the resulting difference in the rates of crystal growth
in both cases mentioned appears due to incorporation of tetramers, see Fig. 14, by the crystal body in the case of crystals based on 2AUB lysozyme.
Figure 14. Growth units: left; l monomeric growth unit of A,B,C and D type, middle; m the lowest energy configuration of four monomeric growth units, right; r generalized
tetrameric growth unit [18]. Realize that in the case of tetrameric growth unit r in contrast to monomeric growth units l every sides have the same hydrophobicity what plays pivotal
role in calculation of the movement probability lack of crossing of any energetic barrier because the geometrically smooth surface is also energetically smoothflat.
A clear origin of emergence of such ordered aggregates could readily be the ordered nanotemplate of LB type from which such entities may easily desorb [97], thus being dis-
persed in the nearby crystal vapor surrounding phase, and being therefore available for the on-nanotemplate-grown crystal drop for a final absorption.
2.2.5. The Computer Model for Protein Crystal Growth by Nanotemplate
