Chem 2440 Project

1
Chem 2440 Project
Xiong, Hui
April, 2003
Extension of Dynamics of Granular Flow
Methodology to Cell Biology
2

• Introduction

Previously, Astarita et al 1 have concluded
that the methodology involved in the dynamics of
polymeric liquid could be extended not only to
granular materials (GM) but possibly to a variety
of areas. Based on their conclusion, A.Kummer
and R. Ocone 2 discussed the logic status of
the steps in GMs microscopic modelling showing
how it can be extended to an apparently
unrelatedarea, i.e. biological system(BS). Since
the existence of cells and their metabolic
processes are undoubtedly demonstrated to exist,
they followed a bottom-up approach contrary to
the procedure by Maxwell and Boltzmann.
3

• Logic of the theories

• Analog Between GM and BS

The first step of the methodology 2 consists
in determining the microstructural element ( of
dimension d) which is the control of the
epigenetic system within the cell 3. The
next step consists in establishing the scale of
description l is chosen to be as the ensemble of
cellular control mechanism 2. Kummer et al
2 then chosen L, the macroscopic scale of the
phenomenon under consideration, coincident with
the cell itself. d needs to satisfy
(1)
4

• Logic of the theories

• Local Morphology

Morphology, in a somewhat loose sense, means
the average state of the microstructural
elements within a neiborhood 2.In GM, Ocone and
Astarita 4 have identified the morphology with
the fluctuating energy or granular
temperature 5,6 the associated average
velocity of the particulate phase, is defined as
? if u is the Instantaneous velocity of a
grain. Consequently, the fluctuating velocity of
the particulate phase is defined as Cu- ?, and
the average kinetic energy per unit mass of the
particles, K, is /2 /2 /2.
Therefore, in addition to the kinetic energy of
the main motion, /2, one has a
fluctuating energy /2. This has been
called the pseudo-temperature 5,6, T, by
analogy with the classical kinetic theory of
gases.
5

• Logic of the theories

• Talandic temperature in the epigenetic system
  (ES)

In the epigenetic system (ES) oscillations are
identified in relation to control mechanisms.
Specific signals, as messenger RNA and enzymatic
proteins, define oscillation within the cells
and for the epigenetic system Goodwin 3
relates oscillations to some kind of
temperature which he called with a talandic
temperature, ?.To understand better the status
of the talandic temperature, Kummer and Ocone
2 considered genetic locus, Li, which
synthesize a messenger RNA (Xi) the latter
signals a ribosome, R, and an enzymatic protein
(Yi) is synthesized which influences the
metabolic state resulting in a metabolite (Mi).
6

• Logic of the theories

• Talandic temperature in the epigenetic system
  (ES)

A fraction of the metabolite loops back to the
genetic locus repressing the protein synthesis
(Fig. 1) 2.
The simple scheme described brings to the
definition of the talandic temperature a
measure of chemical osscillations induced by the
feedback mechanism in individual cells. The
talandic temperature bears only loose analogy
with the true (thermodynamic) temperature and it
has not the dimension of a thermodynamic
temperature ( in the same way as granular
temperature does not).
7

• Logic of the theories

• Calculation of the stress tensor in GM

• , in a GM is still that of compressible Newtonian
  liquid 1

(2)
where D is the particle phase rate of strain
tensor, DD is the deviatoric part of D D/Dt is
the total derivative with respect to the particle
phase motion, p is the equilibrium particulate
pressure, ? is the bulk viscosity and ? is the
shear viscosity. Eq. (2) is a constitutive law
for the internal stress tensor and emerges from
the statistical theory as applied to GM however,
it can be used only if the appropriate
constitutive relations for p, ? and ? are
known2. This implies to know the local
morphology since p, ? and ? depend on the value
of the pseudo temperature, T.
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• Logic of the theories

• Force in ES

Goodwins treatment of the statistical
mechanics of the epigenetic system and the
expressions developed to show the relationship
between the talandic temperature of a cell, ?,
and its competence to response to certain
stimuli are considered 2 . A change in an
environmental parameter will result in a
change of the system in the manner determined by
the stimulus. These changes, connected to the
change in the system free energy, are associated
to a generalized force conjugated to the
external parameter sr 3
(3)
The parameters, external to the single
epigenetic system, affect the system through ci
and bi, which represent the effect of the
neighbourhood. G is the talandic internal
energy associated to the momentum xi and the
position yi. The neighbourhood will cause changes
in the talandic function (e.g. talandic
internal energy) and on the talandic temperaure.
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• Logic of the theories

• Analog in GM and ES

The force in ES shown previously is the perfect
analog for ES of the stress tensor in GM and
therefore it is a constitutive equation for the
interactive force2. Eqs. (2) and (3) both
include the local morphology. In GM, both ? and ?
depend on the value of the granular temperature
similarly in ES, G, the internal energy, is a
function of the talandic temperature, therefore
this means the force is a function of ?. Since
the equation for the internal force in ES and the
stress tensor in GM include the local
morphology, an evolution equation for the
morphology is needed 2. In GM, its a
balance on internal pseudo-energy, U, which is
written as
(4)
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• Logic of the theories

• Analog in GM and ES (Continued)

Pseudo-energy is generated at a rate w per unit
volume and time by the work done by the particle
phase stress tensor2. Pseudo-energy may be
flowing at rate q into the neighbourhood of a
point within the particulate phase and
fluctuating energy can be supplied or generated
at rate Q. I represents the dissi- pation of
energy due to the inelastic collision between
grains. In BS, it is far more complicated to
write the equivalent of Eq. (4). However, one
can still extract the close analogy. The analog
to the equation of motion derived by Newton to
describe the motion of a particle is written as
3
(5)
The above equations are in the Hamiltonian form
and thus the Hamiltonian function G is
automatically obtained. This will make it
possible to construct a statistical mechanics
theory.
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• Thermodynamics

In developing the theory of granular
thermodynamics, Ocone and Astarita 4 clarified
that no quasi-static argument could be applied
and this automatically brings the consideration
that granular entropy to be defined in an
axiomatic way. Consider in Eqn. (4), the presence
of I renders impossible an equilibrium
theory. Consequently, since no equilibrium form
of psedu-thermodynamic theory is possible, no
quasi-static processes are allowed as well 2.
Since the presence of a finite dissipation term
in Eq. (4), no matter how small such term is,
energy is no longer equipartitioned 7. In ES,
the condition of equipartition of energy is
satisfied only under the hypothesis of weak
interaction and steady state, and this shows how
a non-equilibrium thermodynamic theory appears to
be a much more appropriate way to describe the
feature of the system2. Following the route
developed by Astarita and Ocone 4, Kummer and
Ocone 2 outlined a thermodynamic theory for ES
and considered dissipation to occur as the
consequence of degradation of nutrient. From Eqn
(3), they obtained that F is dependent on the
talandic temperature ?, external parameter sr and
their grediant. They then considered a class of
constitutive equations where ?, grad ?, sr and
grad sr , are the local state of the system.
Further, they introduced the talandic partition
function Q
(7)
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• Thermodynamics

The first law of thermodynamic for ES is then a
balance of U2
(8)
where w is the talandic work done as a
consequence of a change of an external parameter
sr, q is the flux of talandic energy and I
comprises all possible dissipation of talandic
energy. In writing the second law for
thermodynamics, kummer and Ocone 2 followed the
approach of Truesdell 8 and regard talandic
entropy, S, as a preimitive quan- tity subject to
the axioms laid down for it. The fist axiom is S
as a function of state
S s(?, grad?, sr, grad sr)
(9) The second axiom is
that the talandic entropy is subjected to the
second law of thermodynamics and therefore the
concept of entropy flux is
(10)
13

• Conclusion

Kummer and Ocone 2 have shown the analog
between the thermodynamic theory for GM 4 and
the epigenetic system, ES 3. In drawing the
analog bet- ween the two classes of systems, they
recognized that the thermodynamic theory
developed for the epigenetic system is indeed not
an equilibrium theory, hence, an axiomatic
treatment 8 is proposed. This constitute the
tools to further develop a non-equilibrium
thermodynamic theory for ES.
14
References
1 T.Astarita, R.Ocone, G. Astarita, J.
Non-Netwtonian Fluid Mech., 76, (1998), 5-25
2 A. Kummer, R. Ocone,Physica A, 321 (2003)
587-597
3 B.C.Goodwin,Temporal Organization in Cells,
Academic Press, New York, 1963
4 R.Ocone, G. Astarita, J. Rheol., 37, (1993),
727-742
5 C.K.K. Lun, S.B. Savage, D.J. Jeffery, N.
Chepurnity, J. Fluid Mech. 140, (1984) 223
6 P.K. Haff, J.Fluid Mech., 134, (1983), 401
7 R.Ocone, G. Astarita, Nature, 375, (1995),
254
8 C.A. Truesdell, Rational Thermodynamics, 2nd
edition, Springer, Berlin, 1984