Entropy Production in a Entropy Production in a System of Coupled Nonlinear System of Coupled Nonlinear

Driven OscillatorsDriven Oscillators

Mladen Martinis, Vesna Mikuta-MartinisRuđer Bošković Institute, Theoretical Physics Division

Zagreb, Croatia

MATH/CHEM/COMP, Dubrovnik-2006

NNonequilibrium onequilibrium thermodynamics of complex thermodynamics of complex

biological networksbiological networks

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

M o t i v a t i o n

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.

4

Equation of balanceEquation of balance

New Old

Out GenIn

= +

- + Con

Δ

Δ -=

Δ = Δe + Δi = balance

Gen = GenerationCon = Consumption

(Δe = in – out) (Δi = Gen – Con)

5

System

Energy balanceEnergy balance

Δ E = Eout - Ein

En

viro

nm

ent

Ein

Eout

6

1st law of thermodynamics1st law of thermodynamics (Energy balance equation)

Energy

Eout

Closed system (no mass transfer)

U internal energy, H = U + PV enthalpyEk kinetic energyEp potential energy ΔQ heat flowΔW workΔWs work to make things flow

ΔE = Eout – Ein = ΔQ – ΔW

E = U + Ek + Ep

ΔE = Eout – Ein = ΔQ – ΔWs

E = H + Ek + Ep

Open system (mass transfer included)

7

System

EnEntropytropy balance balance

ΔeS = Sout - Sin

ΔS = ΔiS + ΔeS

En

viro

nm

ent

ΔiS ≥ 0

SSinin

SSoutout

ΔiS entropy production(EP)

MaxEP MinEP EP ?

8

2nd law of thermodynamics2nd 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 oscillatorsCoupled 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.

10

BioloBiologicalgical oscil oscilllatoratorss

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.

11

Biological networkBiological network

1122

33 4455

66

11

SS

7777

33

22

11

66

55

44

Graph theoryGraph theory

g g 3434

Graph with weighted edgesGraph with weighted edges →→ network network

Network theoryNetwork theory

GraphGraph

G = {gij} connectivity matrix

12

BiologicalBiological hhomeostasisomeostasis(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 ?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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NNegative feedback control egative feedback control stabilizes the systemstabilizes the system

(It is a nonlinear process)

Bio-factor

Receptor Effector

Bio-factor

Receptor Effector

message

Bf increses

No change in Bf

Bf decreases

message

Correctiveresponse

Correctiveresponse

Oscillations around equilibrium

Osmoregulation, Sugar in the blood regulation, Body temperature regulation

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Coupled Coupled n nononlinear oscillinear oscilllatoratorss

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 ωi are randomly distributed across the population with some probability density function (pdf).

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Coupled Coupled n nononlinear oscillinear oscilllatoratorss

• Given natural frquencies ωi

• Given couplings gij

ωi, gij Phase transition

(Kuramoto model)

SynchronizationSelf-organization

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Coupled nonlinear oscillators

ii

kk

dxdxkk/dt = F/dt = Fkk(x(xkk, c, ckk, t) + , t) + ΣΣ g gikik (x (xii, x, xkk))

xxkk = the state vector of an oscillator = the state vector of an oscillator

xxkk = (x = (x1k1k, x, x2k2k), k = 1,2, ..., N), k = 1,2, ..., N

ggikik = coupling function = coupling function

FFkk = intradynamics of an oscillator = intradynamics of an oscillator

Diffusion coupling: gDiffusion coupling: gikik = = μμ ikik (x (xii – x – xkk); ); μμ ikik = NxN matrix = NxN matrix

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Complex phase space Standard phase space:

s(t), sn = biological signal dts(t), sn+1 = rate of change dts(t) = f(s,t) or sn+1 = f(sn)

Free oscillator dt

2s + ω2s = 0, s(t) = Acos( ωt + φ )

Complex phase space:

z(t) = ωs(t) – i dts(t)

dtz(t) = F(z, z*, t) or zn+1 = F(zn, z*n) Free oscillator: dtz = i ω z , |z|2 = const z = r e iθ , dtr = 0, dt θ = ω ωs = Re z = r cosθ

dtssn+1

s,sn

z - planez

r

θ

Re z = r cosθIm z = r sinθ

Re z

Im z

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Limit Cycle Oscillator(LCO)negative feedback effect

Example of LCOExample of LCO:

dtz = (a2 + iω - |z|2)z

dtr = (a2 – r2)r, dt θ = ω ωs(t) = r(t)cos θ(t)

SSolution: r(t) =a/u(t), θ(t) = ωt + θ0

u(t) = [1 – (1- u02)exp(-2a2t)]½

ωs(t) = (a/u(t))cos( ωt + θ0), ω = 2π / T

r0>a

r0<a

a

Limit cycle

z-plane

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Limit cycle property

Limit cycle oscillator

0,0

0,5

1,0

1,5

2,0

2,5

0 0,5 1 1,5 2 3 4 5

t - time

r(t)

r1(t)

r2(t)

r0 > 1

r0 < 1 1

Limit cycle line

r(t) = [1 – (1- r0 -2 )exp(-2t)]-½

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OscillatingOscillating signal signal

-1,5

-1

-0,5

0

0,5

1

1,5

0 5 10 15 20 25 30

time in hours

r(t)

co

sθ(

t)

Circadian signalr0 < 1

ωs(t) = r(t)cosθ(t), ω = π /12

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Limit cycle in the z - plane

Limit cycle

-1,5

-1

-0,5

0

0,5

1

1,5

-1,5 -1 -0,5 0 0,5 1 1,5

z - plane

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Entropy productionEntropy 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, θ)-phase space

dtr(t) = (a2 – r2)r + ς(t), dtθ(t) = ω ς(t) Gaussian white noise < ς(t) ς(t’)> = 2dδ(t – t’)

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Non- equilibrium entropyNon- equilibrium entropy

dtSe

dtSi

dtS = dtSi + dtSe ≥ 0

Si(t) = - ∫rdr p(r,t) lnp(r,t) ≡ <si(t)>

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 equationwith a given initial condition p(r,0) = p0(r)

∂tp(r,t) = - ∂rj(r,t) = - ∂r[(a2 – r2)r - D∂r]p(r,t)

ApplicationsApplications

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). 

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 rhythmusModelling circadian rhythmusas coupled oscillators

1. Blood pressure circadian

2. Heart rate circadian

3. Body temperature circadian

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Three coupled oscillators

BT

BPHR

21

3

21

3

3

21

A B C

D

g12 g12

g23

g12

g23g31

333231

232221

131211

ggg

ggg

ggg

g Coupling matrixgkk = 0gjk ≠ gkj

Single oscillator : dt2x3(t) = - ω3

2 x3 + ...

External stimuli

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Coupled Limit Cycle Oscillators

BT

BPHR

dtzk(t) = (ak + iωk - |zk(t)|2)zk(t) + Σgkj (zj(t) – zk(t)) - i Fkext(t)

g12

g23g31

k = HR, BP, BTgkj = - gjk , gkj = Kk δkj

Kk ≥ 0 coupling strength

zk(t) = rk(t) e iθk(t)

Linear coupling model*

There are six (6) first order differential equations to be solvedfor a given initial conditions (rk(0), θk(0); k = 1, 2, 3)

Fkext

*Aronson et al., Physica D41 (1990) 403

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ConsequencesConsequences

• 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 = |ωk – ωj|.

• Weak coupling (K ~ 0): the oscillators behave as independent units , subjected each to the influence of the external stimuli (Fext (t)).

• With increasing coupling (K> 1) two important classes of stationary solutions are possible:

– The amplitude death (r1, r2 or r3 → 0 as t → ∞ )

– 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 = |ωk – ωj|.

• 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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ENDEND

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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).

.

37

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.

40

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.

41

Transport phenomena(an elementary approach)

jX = ρXv

ρX = X/V density

V = S·L volume

L = v·t

jXS = X/t

X = (mass, energy,

momentum, charge, ...)

S

jX

L

v

X

Current density (flux):

42

Transport phenomena(an elementary approach)

• Continuity equation

∂tρX + div jX = 0• Transport equation

jX = - αX grad ρX

αX(from kinetic theory) ~ vℓ ℓ - mean free path

43

Transport phenomena(kinetic approach)

• The net flux through the middle plane in one direction is

j = (j2 – j1)/6

= - α gradρ

α = vℓ/6

ℓ

ℓ

j2 = vρ(r - ℓ)

j1 = vρ(r + ℓ)

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Transport phenomenaMass, momentum, and energy transport

Diffusion(mass transport)

jD = v[C(x - ℓ) –C(x + ℓ)] / 6

= v( - 2 ℓ ∂x C(x)) / 6

jjDD = - D ∂ = - D ∂xxC(x)C(x)

D = v D = v ℓℓ / 3 / 3ℓℓ

x

C(x - ℓ) C(x + ℓ)

C - concentration

Cv/6

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Transport phenomenaMass, momentum, and energy transport

Heat transver (energy transport)

q = C v[Ek(x - ℓ) –Ek(x + ℓ)] / 6

= C v( - 2 ℓ ∂x Ek(x))/ 6

q = - q = - κκ ∂ ∂xxT(x)T(x)

κκ = v C = v C ℓℓ c / 3 c / 3ℓℓ

x

T(x - ℓ) T(x + ℓ)

C ( concentration ) = N / Vc = ∂E/∂T = specific heat

Cv/ 6

46

Transport phenomenaMass, momentum, and energy transport

Viscosity (momentum transport)

Πxy = C vm[vy(x - ℓ) –vy(x + ℓ)] / 6

= C vm( - 2 ℓ ∂x vy(x)) / 6

Πxy = - = - ηη ∂∂xxvvyy(x)(x)

ηη = Cvm = Cvm ℓℓ / 3 / 3ℓℓ

x

vy(x - ℓ) vy(x + ℓ)

C - concentration

y

Cv/6

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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 = Σ Jα Fα

• 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.