PHYS3410
BIOPHYSICS
II
Lecture Notes
Section
I:
Thermodynamics of Biological Systems
Lecture
9 :Far from equilibrium: new order
Boltzmann
vs Darwin
So, how
can Boltzmann’s type disappearance of unusual more organised
states be reconciled with Darwin’s statistical selection of
rare events? Darwin’s theory begins with an assumption of
the spontaneous fluctuations of species, which are reinforced
and lead to self-organisation and more complexity (more order).
The answer
lies in recognizing three different types of thermodynamics
systems.
The three
stages of thermodynamics
1)
Equilibrium reversible idealised systems: can be described
by Newtonian classical mechanics. Such systems do not evolve
with time. (“The clockwork universe”).
2)
Near equilibrium linear systems: such systems evolve
with time toward equilibrium under constraints of maximization
of entropy (S) and minimization of internal energy (U) and
entropy production (diS/dt). The progress toward
equilibrium may be a flow of heat, flow of matter or a chemical
reaction. The flows Ji are proportional
to their conjugate force Fi, as well as
to the other forces that may exist in the system. At
equilibrium both flows and forces disappear. The entropy
production can be generalised:
 (9.1)
The systems tend to a minimum entropy production, where
dS = 0, (deS = diS): stationary state.
The system exports entropy to its environment. Living
systems often exist near equilibrium.
3)
Far from equilibrium systems: the flows are no longer
proportional to their forces. Very complex behaviour results,
with instabilities, fluctuations and sometime evolution of
new order.
Systems,
which are subject to a flow of energy and matter, can be driven
far from thermodynamic equilibrium, into "nonlinear"
regime: the flows are no longer linear function of the thermodynamic
forces in the system. In case of chemical reactions, the linear
regime occurs if the Affinities are small compared to RT (~2.5
kJ/mole at T = 300 K). Since Affinities can easily reach the
range 10 -100 kJ/mole, the non-linear regime is not uncommon
with chemical reactions. Heat conduction and diffusion approach
such conditions more seldom.
However,
in nature, far from equilibrium systems are ubiquitous. The
earth is an open system subject to the constant flow of energy
from the sun. This influx of energy provides the driving force
for the maintenance of out-of -equilibrium state, which makes
life possible. Every living cell survives through the flow
of matter and energy.
For the
far from equilibrium systems, there are no general extremum
principles that predict the state to which it will evolve.
Thus, these systems can evolve unpredictably, their state
cannot be always uniquely specified by macroscopic rate equations.
This is because, for a given set of non-equilibrium conditions,
it is often possible to have more than one state. As a result
of random fluctuations, or other random factors, the system
evolves to one of the many possible states. These new states
can be highly organised states.
Thus
the irreversible dissipative processes can be both creators
of disorder and of new order.
Far from
equilibrium a new structures come into being: dissipative
structures.
Parameters
describing crystal structures derive from the properties of
molecules from which they are composed. Dissipative structures
are a reflection of the global situation of non-equilibrium
producing them. The parameters are macroscopic and time scales
do not correspond to molecular times.
Bifurcation
and Symmetry Breaking
At equilibrium
and near equilibrium there is only one steady state, which
depends on the values of some control parameters. In far from
equilibrium, there may be several steady states. Which of
these will result?
In some
cases this depends on the history of the system. For instance,
define a parameter l
(which may be a concentration of a substance). A slow increase
of l will result in different behaviour to starting
point with large concentration X (Fig.a). In some cases two
branches are equally probable (Fig. b) and a random element
appears. The macroscopic equations cannot predict which path
the system will take. If the both branches are equally probable,
we expect to find (after many repetitions of the experiment)
half of the systems in the upper state and half in the lower
state.
However,
in the world around us, some simple symmetries are broken:
the chemicals involved in living systems display remarkable
asymmetry! A molecule whose geometrical structure is not identical
to its mirror image is said to possess chirality (handedness).
The mirror image structures of a chiral molecule are called
L- and D- enantiomers (L “levo” and D for “dextro”). Amino
acids, building blocks of proteins, and deoxyribose in DNA
are chiral molecules. From bacteria to man, nearly all amino
acids that take part in the chemistry of life are L-amino
acids. The riboses in DNA
and RNA are D-riboses. This is all more remarkable, as
chemical reactions show equal preference for the mirror image
forms. While we can envisage sets of equations, which will
model chiral asymmetry in chemical reactions, the origin of
life’s homochirality remains a mystery: did it arise in pre-biotic
era and facilitated evolution of living organisms, or did
primitive form of life incorporate both L- and D- substances
and later one branch overtook the other?
Cascading
bifurcations
The primary
bifurcation often pushes the system beyond the threshold of
stability. The system might find more bifurcation points and
many possible states. Oscillations are observed with a period
T. This period doubles as the control parameter is increased.
Examples
Benard
cell
Vertical
temperature gradient set up in horizontal liquid layer. The
lower surface of the liquid is heated. A heat flux is set
up. At some critical threshold value, the fluid will start
to move (convection). The convection produces complex spatial
organization of hexagonal patterns.
Turbulence
Orderly,
laminar flow of a fluid will occur when certain flow rate
is exceeded.
Chaos
and self organisation in biological systems
The formation
of lipid bilayers probably led to evolution of first cells.
The aggregation
of slime mold Dictyostelium discoideum
The segregation
of Ventricaria ventricosa.
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