Entropy Production in a System of Coupled Nonlinear Driven Oscillators

1
Entropy Production in a System of Coupled
Nonlinear Driven Oscillators

• Mladen Martinis, Vesna Mikuta-Martinis
• Ruder Boškovic Institute, Theoretical Physics
  Division
• Zagreb, Croatia

MATH/CHEM/COMP, Dubrovnik-2006
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Nonequilibrium thermodynamics of complex
biological networks
M o t i v a t i o n

• What are the thermodynamic links between
  biosphere and environment?
• How to bring nonequilibrium thermodynamics to the
  same level of clarity and usefulness as
  equilibrium thermodynamics?
• Energy balance analysis
• Entropy production as a measure of Bio ??
  Env interaction

3

• "A violent order is disorder and a great
• disorder is an order. These two things
• are one.
• Wallace
  Stevens, Connoisseur of Chaos, 1942

• Non-equilibrium may be a source of order
• Irrevesible processes may lead to disspative
  structures
• Order is a result of far-from-equilibrium
  (dissipative)
• systems trying to maximise stress reduction.

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Equation of balance
Old
New
?


(?e in out)
(?i Gen Con)
Out
Gen
In
Con
?
-


-
? ?e ?i balance
Gen Generation Con Consumption
5
Ein
Energy balance ? E Eout - Ein
System
Environment
Eout
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1st law of thermodynamics (Energy balance
equation)
Closed system (no mass transfer)

• ?E Eout Ein ?Q ?W
• E U Ek Ep

U ? internal energy, H U PV ? enthalpy Ek ?
kinetic energy Ep ? potential energy ?Q ? heat
flow ?W ? work ?Ws ? work to make things flow
Open system (mass transfer included)
?E Eout Ein ?Q ?Ws E H Ek Ep
Eout
Energy
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Entropy balance ?eS Sout - Sin ?S ?iS
?eS
Sin
System
Environment
?iS 0
?iS ? entropy production(EP)
Sout
MaxEP MinEP
EP ?
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2nd law of thermodynamics

• Entropy production (diS/dt)
• dS deS diS with diS 0
• Entropy production includes many effects
  dissipation, mixing, heat transfer, chemical
  reactions,...

9
Coupled oscillators

• Many (quasi)periodic phenomena in physics,
  chemistry, biology and engineering can be
  described by a network of coupled oscillators.
• The dynamics of the individual oscillator in the
• network, can be either regular or
  complicated.
• The collective behavior of all the oscillators in
  the network can be extremely rich, ranging from
  steady state (periodic oscillations) to chaotic
  or turbulent motions.

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Biological oscillators

• It is well known that cells, tissues and
• organs behave as nonlinear oscillators.
• By the evolution of the organism,they are
• multiple hierarchicaly and functionally
• interconnected
• ? complex biological network.

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Biological network
Graph
1
4
1
1
Graph theory
2
2
5
3
4
g 34
5
6
3
6
S
Graph with weighted edges ? network
Network theory
7
7
G gij connectivity matrix
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Biological homeostasis(Dynamic self-regulation)

• Homeostasis (resistance to change) is the
  property of an open system, (e.g. living
  organisms), to regulate its internal
  physiological environment
• to maintain its stability under external varying
  conditions,
• by means of multiple dynamic equilibrium
  adjustments, controlled by interrelated negative
  feedback regulation mechanisms.
• Most physiological functions are mainteined
  within relatively narrow limits
• ( ? state of physiological homeostasis).

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What is feedback ?

• It is a connection between the output of a system
  and its input
• ( ? effect is fed back to cause ).
• Feedback can be
• negative (tending to stabilise the system ?
  order) or
• positive (leading to instability ? chaos).
• Feedback results in nonlinearities
• leading to unpredictability.

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Negative feedback control stabilizes the
system(It is a nonlinear process)
message
Receptor
Effector
Corrective response
Bf increses
No change in Bf
Bio-factor
Bio-factor
Oscillations around equilibrium
Bf decreases
Corrective response
message
Receptor
Effector
Osmoregulation, Sugar in the blood regulation,
Body temperature regulation
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Coupled nonlinear oscillators

• Each oscillating unit (cell, tissue, organ, ...)
  is modelled as a nonlinear oscillator with a
  globally attracting limit cycle (LC).
• The oscillators are weakly coupled gij , and
  their natural frequencies ?i are randomly
  distributed across the population with some
  probability density function (pdf).

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Coupled nonlinear oscillators
(Kuramoto model)

• Given natural frquencies ?i
• Given couplings gij

?
Phase transition
?i, gij
Synchronization Self-organization
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Coupled nonlinear oscillators
dxk/dt Fk(xk, ck, t) S gik (xi, xk)
xk the state vector of an oscillator xk (x1k,
x2k), k 1,2, ..., N gik coupling function Fk
intradynamics of an oscillator
i
k
Diffusion coupling gik µ ik (xi xk) µ ik
NxN matrix
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Complex phase space

• Standard phase space
• s(t), sn biological signal
• dts(t), sn1 rate of change
• dts(t) f(s,t) or sn1 f(sn)
• Free oscillator
• dt2s ?2s 0, s(t) Acos( ?t f )
• Complex phase space
• z(t) ?s(t) i dts(t)
• dtz(t) F(z, z, t) or zn1 F(zn, zn)
• Free oscillator dtz i ? z , z2 const
• z r e i? , dtr
  0, dt ? ?
• ?s Re z r cos?

dts sn1
s,sn
Im z
z - plane
z
Re z r cos? Im z r sin?
r
?
Re z
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Limit Cycle Oscillator(LCO)negative feedback
effect
r0gta

• Example of LCO
• dtz (a2 i? - z2)z
• dtr (a2 r2)r, dt ? ?
• ?s(t) r(t)cos ?(t)
• Solution r(t) a/u(t), ?(t) ?t ?0
• u(t) 1 (1- u02)exp(-2a2t)½
• ?s(t) (a/u(t))cos( ?t ?0), ? 2p / T

z-plane
r0lta
a
Limit cycle
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Limit cycle property
r(t) 1 (1- r0 -2 )exp(-2t)-½
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Oscillating signal
?s(t) r(t)cos?(t), ? p /12
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Limit cycle in the z - plane
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Entropy productionin a driven LC oscillator

• Biological systems are generically out of
  equilibrium.
• In an environment with constant temperature the
  source of non-equilibrium are usually mechanical
  (external forces) or chemical
• (imbalanced reactions) stimuli with
  stochastic character of the non-equilibrium
  processes.

Stochastic (Langevin) description of a driven LC
oscillator representing stochastic trajectory
(dts(t), s(t))) in (r, ?)-phase space
dtr(t) (a2 r2)r ?(t), dt?(t) ?
?(t) ? Gaussian white noise lt ?(t) ?(t)gt
2dd(t t)
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Non- equilibrium entropy
Si(t) - ?rdr p(r,t) lnp(r,t) ltsi(t)gt si(t)
- lnp(r,t), dtse(t) dtq(t)/ T (a2 r2)r
dtr, D T p(r,t) is the probability to find
the LCO in the state r p(r,t) is the solution of
the the Fokker-Planck equation with a given
initial condition p(r,0) p0(r)
dtSe
dtSi
dtS dtSi dtSe 0
?tp(r,t) - ?rj(r,t) - ?r(a2 r2)r -
D?rp(r,t)
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Applications
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Biological Rhythms (BRs)

• BRs are observed at all levels of living
• organisms.
• BRs can occur daily, monthly,
• or   seasonally. 
• Circadian (daily) rhythms (CRs)
• vary in length from species to species
• (usually lasts approximately 24 hours). 

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Biological Clocks (BCs)

• Biological clocks are responsible for
• maintaining circadian rhythms, which
  affect
• our sleep, performance, mood and more.
• Circadian clocks enhance the fitness of an
• organism by improving its ability to adapt to
• environmental influences, specifically daily
• changes in light, temperature and humidity.
•  

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Modelling circadian rhythmusas coupled
oscillators

• Blood pressure circadian
• Heart rate circadian
• Body temperature circadian

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Three coupled oscillators
Single oscillator dt2x3(t) - ?32 x3 ...
3
BT
3
External stimuli
g23
B
C
BP
A
2
2
1
1
HR
g12
g12
3
g23
g31
Coupling matrix gkk 0 gjk ? gkj
D
2
1
g12
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Coupled Limit Cycle Oscillators
Linear coupling model
Aronson et al., Physica D41 (1990) 403

• BT

g23
g31
k HR, BP, BT gkj - gjk , gkj Kk dkj Kk
0 coupling strength zk(t) rk(t) e i?k(t)
BP
HR
g12
Fkext
dtzk(t) (ak i?k - zk(t)2)zk(t) Sgkj
(zj(t) zk(t)) - i Fkext(t)
There are six (6) first order differential
equations to be solved for a given initial
conditions (rk(0), ?k(0) k 1, 2, 3)
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Consequences

• Coupled limit cycle oscillator model has variety
  of stationary and nonstationary solutions which
  depend on the coupling K, the limit cycle radius
  a and the frequency differencies ?kj ?k ?j.
• Weak coupling (K 0) the oscillators behave as
  independent units , subjected each to the
  influence of the external stimuli (Fext (t)).
• With increasing coupling (Kgt 1) two important
  classes of stationary solutions are possible
• The amplitude death (r1, r2 or r3 ? 0 as t ? 8
  )
• The frequency locking (synchronization)

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Conclusion

• We have developed a mathematical models of BP, HR
  and BT circadian oscillations using the coupled
  LC oscillators approach.
• Coupled LC oscillator-model can have variety of
  stationary and nonstationary solutions which
  depend on the coupling K, the limit cycle radius
  a and the frequency differencies ?kj ?k ?j.
• Weakly coupled oscillators behave as independent
  units but with coupled phases.
• They are subjected each to the influence of the
  external disturbancies (Fext (t)) which can
  change circadian organization of the organism and
  become an important cause of morbidity.

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Self-organization

• Self-organization in biological systems
• relies on functional interactions between
• populations of structural units (molecules,
  cells, tissues, organs, or organisms).
• .

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Synchronization

• There are several types of synchronization
• Phase synchronization (PS),
• Lag synchronization (LS),
• Complete synchronization (CS), and
• Generalized synchronization (GS)
• (usually observed in coupled chaotic
  systems)

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Relationship between entropy and self-organization

• The relationship between entropy and
  self-organization tries to relate organization to
  the 2nd Law of Thermodynamics ? order is a
  necessary result of far-from-equilibrium
  (dissipative) systems trying to maximise stress
  reduction. This suggests that the more complex
  the organism then the more efficient it is at
  dissipating potentials, a field of study
  sometimes called 'autocatakinetics' and related
  to what has been called 'The Law of Maximum
  Entropy Production'. Thus organization does not
  'violate' the 2nd Law (as often claimed) but
  seems to be a direct result of it.

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What are dissipative systems ?

• Systems that use energy flow to maintain their
  form are said to be dissipative (e.g. living
  systems ).
• Such systems are generally open to their
  environment.

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Biological signals

• Every living cell, organ, or organism generates
• signals for internal and external
  communication.
• In-out relationship is generated by a
  biological
• process (electrochemical, mechanical,
  biochemical
• or hormonal).
• The received signal is usually very distorted
  by the
• transmission channel in the body.

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Transport phenomena(an elementary approach)

• jX ?Xv
• ?X X/V density
• V SL volume
• L vt
• jXS X/t
• X (mass, energy,
• momentum, charge, ...)

Current density (flux)
v
S
X
L
jX
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Transport phenomena(an elementary approach)

• Continuity equation
• ?t?X div jX 0
• Transport equation
• jX - aX grad ?X
• aX(from kinetic theory) vl
• l - mean free path

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Transport phenomena(kinetic approach)

• The net flux through the middle plane in one
  direction is
• j (j2 j1)/6
• - a grad?
• a vl/6

j1 v?(r l)
l
l
j2 v?(r - l)
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Transport phenomenaMass, momentum, and energy
transport

• Diffusion(mass transport)

C(x - l)
C(x l)
jD vC(x - l) C(x l) / 6 v( - 2 l
?x C(x)) / 6 jD - D ?xC(x) D v l / 3
Cv/6

l
l
x
C - concentration
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Transport phenomenaMass, momentum, and energy
transport

• Heat transver (energy transport)

T(x - l)
T(x l)
q C vEk(x - l) Ek(x l) / 6 C v( - 2
l ?x Ek(x))/ 6 q - ? ?xT(x) ? v C l c / 3
Cv/ 6

l
l
x
C ( concentration ) N / V
c ?E/?T specific heat
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Transport phenomenaMass, momentum, and energy
transport

• Viscosity (momentum transport)

vy(x - l)
vy(x l)
?xy C vmvy(x - l) vy(x l) / 6 C vm(
- 2 l ?x vy(x)) / 6 ?xy - ? ?xvy(x) ? Cvm l
/ 3
y
Cv/6

l
l
x
C - concentration
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• ENTROPY PRODUCTION
• At the very core of the second law of
  thermodynamics we find the basic distinction
• between reversible and irreversible processes
  (1). This leads ultimately
• to the introduction of entropy S and the
  formulation of the second
• law of thermodynamics. The classical formulation
  due to Clausius refers to
• isolated systems exchanging neither energy nor
  matter with the outside world.
• The second law then merely ascertains the
  existence of a function, the entropy
• S, which increases monotonically until it reaches
  its maximum at the state of
• thermodynamic equilibrium,
• (2.1)
• It is easy to extend this formulation to systems
  which exchange energy and
• matter with the outside world. (see fig. 2.1).
• Fig. 2.1. The exchange of entropy between the
  outside and the inside.

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• To extend thermodynamics to non-equilibrium
  processes we need an explicit
• expression for the entropy production.
• Progress has been achieved along this
• line by supposing that even outside equilibrium
  entropy depends only on the
• same variables as at equilibrium. This is the
  assumption of local equilibrium
• (2). Once this assumption is accepted we obtain
  for P, the entropy
• production per unit time,
• (2.3) dtSi S Ja Fa
• where the Jp are the rates of the various
  irreversible processes involved (chemical
• reactions, heat flow, diffusion. . .) and the F
  the corresponding generalized
• 266 Chemistry 1977
• forces (affinities, gradients of temperature, of
  chemical potentials . . .). This
• is the basic formula of macroscopic
  thermodynamics of irreversible processes.