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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
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NNonequilibrium onequilibrium thermodynamics of complex thermodynamics of complex
biological networksbiological networksWhat 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
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"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 balanceEquation of balance
New Old
Out GenIn
= +
- + Con
Δ
Δ -=
Δ = Δe + Δi = balanceGen = GenerationCon = Consumption
(Δe = in – out) (Δi = Gen – Con)
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System
Energy balanceEnergy balanceΔ E = Eout - Ein
Envi
ronm
ent
Ein
Eout
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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 – ΔWE = U + Ek + Ep
ΔE = Eout – Ein = ΔQ – ΔWs
E = H + Ek + Ep
Open system (mass transfer included)
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System
EnEntropytropy balance balanceΔeS = Sout - Sin
ΔS = ΔiS + ΔeS
Envi
ronm
ent
ΔiS ≥ 0
SSinin
SSoutout
ΔiS entropy production(EP)
MaxEP MinEP EP ?
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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,...
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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.
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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.
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Biological networkBiological network
1122
33 4455
66
11
SS
77 77
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
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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 oscillatorxxkk = (x = (x1k1k, x, x2k2k), k = 1,2, ..., N), k = 1,2, ..., Nggikik = coupling function = coupling functionFFkk = 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)zdtr = (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<aa
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)c
osθ(
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)
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ApplicationsApplications
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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 rhythmusModelling circadian rhythmusas coupled oscillators
1. Blood pressure circadian2. Heart rate circadian3. 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
ggggggggg
g Coupling matrixgkk = 0gjk ≠ gkj
Single oscillator : dt2x3(t) = - ω3
2 x3 + ...
External stimuli
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Coupled Limit Cycle OscillatorsBT
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). .
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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 = 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):
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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
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Transport phenomena(kinetic approach)
• The net flux through the middle plane in one direction is
j = (j2 – j1)/6
= - α gradρ α = vℓ/6
ℓℓ
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
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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
yCv/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.