BioSystems 94 (2008) 242–247
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On the spherical prototype of a complex dissipative late-stage
formation seen in terms of least action Vojta–Natanson principle
Department of Modeling of Physicochemical Processes, Institute of Mathematics & Physics,
University of Technology & Life Sciences, PL-85796 Bydgoszcz, Poland
Received 23 April 2008
Accepted 1 June 2008
This work is dedicated to Prof. Zbigniew J.
Grzywna, Silesian University of Technology
(SUT) in Gliwice, on the occasion of his
special role played during the two-day
Symposium on Bionanomaterials with
Structure–Property Relationship, 10–11
March 2008, SUT, Gliwice, Poland.
Late-stage crystal/aggregate formation
Least dissipative dynamics
The spherical prototype of a crystalline and/or disorderly formation may help in understanding the ﬁnal
stages of many complex biomolecular arrangements. These stages are important for both naturally orga-
nized simple biosystems, such as protein (or, other amphiphilic) aggregates in vivo, as well as certain their
artiﬁcial counterparts, mimicking either in vitro or in silico their structure–property principal relationship.
For our particular one-seed based realization of a protein crystal/aggregate late-stage nucleus grown from
nearby ﬂuctuating environment, it turns out that the (osmotic-type) pressure could be, due to local inho-
mogeneities, and their dynamics shown up in the double layer tightly surrounding the growing object,
still an appreciably detectable quantity. This is due to the fact that a special-type generalized thermo-
dynamic (Vojta–Natanson) momentum, subjected to the nucleus’ surface, is manifested interchangeably,
whereas the total energy of the solution in the double layer could not be such within the stationary regime
explored. It is plausible since the double layer width, related to the object’s surface, contributes ultimately,
while based on the so-deﬁned momentum’s changes, to the pressure within this narrow ﬂickering zone,
while leaving the total energy fairly unchanged. From the hydrodynamic point of view, the system behaves
quite trivially, since the circumventing ﬂow should rather be of laminar, thus not-with-matter supplying,
© 2008 Elsevier Ireland Ltd. All rights reserved.
In this work, we are going to convince the reader that an
interface-controlled sphere growth in a ﬂuctuating environment
(Siódmiak et al., 2007), which may be seen as a crude though still
reliable prototype of growing a (complex) crystal, or (bio)molecular
aggregate, in late-stage stochastic conditions, may be ruled ulti-
mately by corresponding least action principle(s), pointing fairly to
conservative and minimum-dissipation dynamics of the formation
at its close-to-equilibrium state.
Thus, in this particular study we wish to stress very much
a dynamic aspect of the, in general, complex formation, over
its purely thermodynamic counterpart that has been developed
elsewhere (Gadomski, 2007). We see a quite appealing need for per-
forming the study of such a type since the dynamic aspects are, as
being often hardly accessible experimentally, postponed or left for
future studies as quite difﬁcult to perform, except that a computer-
simulation oriented stream of research (Bratko and Blanch, 2001)
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from time to time attempts on uncovering some interesting aspects
of the complex phenomena—realize that crystalline protein aggre-
gations are very often accomplished in practice by trial-and-error
“method”, or quite equivalently, by tedious scanning over multi-
parametric windows of data (Haas and Drenth, 1998).
Notice that such aggregations also appear unavoidably as
by-products of spurious physiological, thus genetic-environmental
conditions—they are described as failures to otherwise non-
problematic functioning of our organisms (Gsponer and
Vendruscolo, 2006)—as an example appearances of neurode-
generative diseases (Parkinson; Alzheimer; Creutzfeldt-Jacob;
etc.), often associated with ﬁbril formations, should be invoked
(Krebs et al., 2005). In the light of the above, a study based on
certain physical, least action concerning principia and (thermody-
namic) rules, might be of appreciable help, and this stands for a
basic motivation, and novelty, of the present work.
Throughout this study, we will use, for transparency and
simplicity reasons, a spherical approximation to otherwise
non-spherical problem—the formation of complex biomolecular
aggregates such as ﬁbrils and/or non-Kossel crystals (Krebs et al.,
2005; Chernov, 1997). The rest of the paper is devoted to explore
this approximation in terms of least action, or equivalently, least
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