Introductory biophysics A. Y. 2017-18 · Reaction-diffusion processes and morphogenesis Edoardo...

Introductory biophysicsA. Y. 2017-18

2. Reaction-diffusion processes and morphogenesis

Edoardo MilottiDipartimento di Fisica, Università di Trieste

Edoardo Milotti - Introductory biophysics - A.Y. 2017-18

Reaction-diffusion equations

∂n∂t

= D ∂2n∂x2

− λn

Example: exponential absorption

x ∈ −a,a[ ]with

The, exact, stationary solution with symmetrical boundary conditions is

n(x, t ! 1) = n0cosh(

p�/Dx)

cosh(p

�/Da

We find the solution by selecting first a Fourier component for the time dependent part

then, by substitution, we find

The last equation is a second order differential equation much like the usual equation for the harmonic oscillator, albeit with a complex coefficient, and we solve it as usual with an exponential function

so that the new parameter is the solution of the equation


@n

@t= D

@2n

@x2� �n

n(x) = n0e↵x

n(x, t) = n(x)e±i!t

D@2n

@x2� (�± i!)n(x) = 0

D↵2 � (�± i!) = 0 ↵ = ±r

�± i!

D


Now we notice that

and we find

Note also that

�± i! =p�2 + !2e±i arctan(!/�)

e±i arctan(!/�)/2 = cosarctan(!/�)

2± i sin

arctan(!/�)

2

=

vuut1

2

1 +

1p1 + !2/�2

!± i

vuut1

2

1� 1p

1 + !2/�2

!

↵ = ±r

�+ i!

D= ±

✓�2 + !2

D2

◆1/4

e±i arctan(!/�)/2


Therefore

and eventually we find the general solution

Where the ± sign corresponds to positive/negative frequencies. The general solution

must satisfy the boundary conditions, and now we let

(two Fourier components for the time-dependent part, m < 1)

n(±a, t) = u0 (1 +m cos!t)

↵ = ±✓�2 + !2

D2

◆1/40

@

vuut1

2

1 +

1p1 + !2/�2

!± i

vuut1

2

1� 1p

1 + !2/�2

!1

A

n(x, t) =

8<

:A exp

2

4✓�2 + !2

D2

◆1/40

@

vuut1

2

1 +

1p1 + !2/�2

!± i

vuut1

2

1� 1p

1 + !2/�2

!1

Ax

3

5

+B exp

2

4�✓�2 + !2

D2

◆1/40

@

vuut1

2

1 +

1p1 + !2/�2

!± i

vuut1

2

1� 1p

1 + !2/�2

!1

Ax

3

5

9=

; e±i!t

Then the DC component corresponds to the partial solution

and therefore

implies

and


n(x, t)|DC = A exp⇣p

�/Dx⌘+B exp

⇣�p

�/Dx⌘

n(a, t)|DC = A exp⇣p

�/Da⌘+B exp

⇣�p

�/Da⌘

= A exp⇣�p�/Da

⌘+B exp

⇣p�/Da

⌘= u0

A = B =u0

2 coshp�/Da

n(x, t)|DC = u0cosh

p�/Dx

coshp

�/Da


The derivation can be replicated for the time-dependent part, and we find

where we let

n(x, t) = u0

cosh

p�/Dx

coshp

�/Da+

m

2

e↵+x + e�↵+x

e↵+a + e�↵+aei!t +

m

2

e↵�x + e�↵�x

e↵�a + e�↵�ae�i!t

!

↵± =

✓�2 + !2

D2

◆1/40

@

vuut1

2

1 +

1p1 + !2/�2

!± i

vuut1

2

1� 1p

1 + !2/�2

!1

A


x/a

n(x,t)

t

D = 0.075l = 0.5w = 12 lm = 0.95a = 1

A pattern with slanting stripes ...


Stripes?


From Anton Seder (German, 1850–1916): Das Thier in der Decorativen Kunst

(Photograph of original print owned by Istituto A. Volta, Trieste)


Ernst Heinrich Philipp August Haeckel

Born: 16 February 1834

Died: 9 August 1919

He was a German biologist, naturalist, philosopher, physician, professor and artist who discovered, described and named thousands of new species, mapped a genealogical tree relating all life forms, and coined many terms in biology, including anthropogeny, ecology, phylum, phylogeny, stem cell, and the kingdom Protista.

Haeckel promoted and popularized Charles Darwin's work in Germany and developed the controversial recapitulation theory ("ontogeny recapitulates phylogeny") claiming that an individual organism's biological development, or ontogeny, parallels and summarizes its species' evolutionary development, or phylogeny. By now recapitulation theory is largely discredited in biology. (above text adapted from Wikipedia).

The published artwork of Haeckel includes over 100 detailed, multi-colour illustrations of animals and sea creatures collected in the book Kunstformen der Natur.


Bacterial colony, from H. Levine and E. Ben-Jacob: “Physical schemata underlying biological pattern formation—examples, issues and strategies”, Phys. Biol. 1 (2004) P14


The Belousov-Zhabotinsky reaction

One of the most common variations on this reaction uses malonic acid (CH2(CO2H)2) as the acid and potassium bromate (KBrO3) as the source of bromine

3CH2(CO2H)2 + 4BrO-3 → 4Br- + 9CO2 + 6H2O


Alan Mathison Turing

Born: 23 June1912Died: 7 June 1954

He was highly influential in the development of computer science, giving a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which can be considered a model of a general purpose computer. Turing is widely considered to be the father of computer science and artificial intelligence.

During World War II, Turing worked for the Government Code and Cypher School (GC&CS) at Bletchley Park, Britain's codebreaking centre. He devised a number of techniques for breaking German ciphers.

In 1948 Turing became interested in mathematical biology. He wrote a paper on the chemical basis of morphogenesis, and predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, which were first observed in the 1960s.


Phil. Trans. R. Soc. Lond. B 1952 237, 37-72


Coupled Turing equations

∂u∂t

= D ∂2u∂x2

− λu + F u,v( )∂v∂t

= D ∂2v∂x2

− λv +G u,v( )

rate of concentration change diffusion

reaction

−λu + F u,v( ) = auu + buv + cu−λv +G u,v( ) = avu + bvv + cv

Turing’s linear polynomial choice


Analogy with the damped harmonic oscillator

d 2xdt 2

= −ω 02x − Γ dx

dt

dxdt

= v

dvdt

= −ω 02x − Γv

linear polynomial structure as in Turing’s choice


Spatial coherence resonance in excitable biochemical mediainduced by internal noise

Marko Gosak, Marko Marhl, Matjaž Perc ⁎

Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia

Received 3 February 2007; received in revised form 18 April 2007; accepted 18 April 2007Available online 24 April 2007

Abstract

We show that in a spatially extended excitable medium, presently modelled with diffusively coupled FitzHugh–Nagumo neurons, internalstochasticity is able to extract a characteristic spatial frequency of waves on the spatial grid. Internal noise is introduced via a stochastic simulationmethod and is the only agent acting on the system. Remarkably, the spatial periodicity is best pronounced at an intermediate level of internalstochasticity. Thus, the reported phenomenon is an observation of internal noise spatial coherence resonance in excitable biochemical media.© 2007 Elsevier B.V. All rights reserved.

Keywords: Excitable biochemical media; Neuronal dynamics; Internal noise; Spatial coherence resonance

1. Introduction

It is a well-known fact that noise can have constructive effectsin different nonlinear dynamical systems. It has been shown that aproper amount of noise can resonantly amplify weak signals,which is known as stochastic resonance [1–3]. Fascinatingly, noisecan play an ordering role even in the absence ofweak deterministicstimuli, whereby the established term describing the phenomenonis coherence resonance [4–7]. Especially in biochemical reactionsystems, it is a rather firmly established fact that a certain degree ofstochasticity is unavoidable. Therefore, many studies have beenperformed for a variety of biological processes, where theconstructive role of noise has been investigated. It has beenshown, that noise can induce stochastic calcium oscillations in asubthreshold system, whereby the best temporal order is obtainedby an intermediate intensity of noise [8–11]. Similar studies havealso been performed for circadian rhythms [12,13] as well asgenetic regulation [14] and neuronal systems [15,16].

Remarkably, noise induced resonance phenomena are notrestricted only to the adjustment of temporal dynamics. Severalstudies are focused also on the spatial or spatiotemporaldynamics of biochemical media [17–19], whereby frequently

the impact of noise on the spatiotemporal order is investigated.It has been discovered that noise alone often suffices to inducespatiotemporally ordered behaviour in systems as diverse asoptical devices and biochemical media [20,21]. In particular,spatiotemporal stochastic resonance has been first reported in[22] for excitable systems. Moreover, there also exist studiesreporting noise-induced spiral growth and enhancement ofspatiotemporal order [23–25], noise sustained coherence ofspace-time clusters and self-organized criticality [26], noiseinduced excitability [27], noise induced propagation ofharmonic signals [28], as well as noise sustained and controlledsynchronization [29] in spatially extended systems.

Recently, specifically the spatial dynamics of noise-inducedexcitatory events in spatially extended systems has beeninvestigated in great detail. Carrillo et al. [30] have shown that,for a nonlinear medium near a pattern-forming instability, thereexists an intermediate value of additive spatiotemporal noise forwhich the peak of the circularly averaged spatial structure functionis best resolved, thus marking spatial coherence resonance in thesystem. Recently, the concept was extended and spatial coherenceresonance phenomena have been reported in excitable media[31,32] and in networks with different topologies [33].

Probably the most famous excitable medium, studied in theframework of physiology, is the neuronal network. It has beenevidenced, that the consistency of spike generation and signal

Biophysical Chemistry 128 (2007) 210–214http://www.elsevier.com/locate/biophyschem

⁎ Corresponding author. Tel.: +386 2 2293643; fax: +386 2 2518180.E-mail address: [email protected] (M. Perc).

0301-4622/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.bpc.2007.04.007


transition in networks of neurons depends on intrinsic noise,which may originate from fluctuations of channel gating,synaptic release, or of background presynaptic activity [34].The role of noise in neuronal networks, and excitable media ingeneral, has been and still is a vibrant avenue of research, thusyielding many experimental and theoretical studies over the pastdecade [35]. While real-life biological neurons exhibit extreme-ly complex behavior, neuronal dynamics must be considerablysimplified in order to make networks computationally traceable.Therefore, the neuronal medium is presently modelled byFitzHugh–Nagumo equations [36,37], which have been derivedfrom the Hodgkin–Huxley model describing the excitabledynamics of electrical signal transmission along neuron axons[38].

Typically, when studying the spatial dynamics of excitablemedia, for example that of neuronal networks, the role ofdifferent intensities of external additive noise has beeninvestigated [32]. Importantly, we presently extend the ideasof previous authors and investigate the spatial order of excitablemedia in dependence on internal noise only, which is introducedby using a stochastic simulation method. We demonstrate that

internal stochasticity is sufficient to extract a characteristicspatial frequency of excitable media, which is best expressed foran intermediate level of internal noise. The phenomenon is thusan observation of internal noise spatial coherence resonance inexcitable media.

2. Mathematical model

As already mentioned above, local units of the excitablemedium under study are governed by the FitzHugh–Nagumoequations [36,37]

dudt

¼ f u ; vð Þ ¼ 1eu 1$ uð Þ u $ vþ b

a

! "; ð1Þ

dvdt

¼ g u ; vð Þ ¼ u $ v: ð2Þ

Individual units are arranged on the L×L (i, j ∈ [1, L])square lattice with no-flux boundary conditions, whereby the

Fig. 1. Characteristic snapshots of the spatial profile of u i,j for χ=(80; 180; 220; 240) increasing from the top left towards the bottom right panel. All snapshots aredepicted on square grids of linear size L=128. The colour mapping is linear, white depicting 0.0 and black 1.0 values of u i,j. For χ=240 the colour scale extends from0.0 to 0.01 in order to reveal small-amplitude fluctuations around the excitable steady state of each unit. Evidently, only for χ=180 the spatial domain is characterizedby periodic excitatory waves, whereas for χ=80 the excitations are randomly scattered. For χ=220 the level of internal stochasticity is too low to induce periodicwaves; only isolated and uncorrelated waves can be noticed, which disappear completely if χ is increased further.

211M. Gosak et al. / Biophysical Chemistry 128 (2007) 210–214

FitzHugh-Nagumo dynamics

+

diffusive correlation term

+

internal noise from Gillespie algorithm

spatial extension is modelled by an additional diffusive flux ofthe form D∇2u i,j that is added to the differential equationdescribing changes of variable u :

du i;jdt

¼ f u i; j; vi; j! "

þ Dj2u i; j; ð3Þ

dvi;jdt

¼ g u i; j; vi; j! "

: ð4Þ

The membrane potential u i, j(t) and time-dependent conduc-tance of potassium channels vi, j(t) are considered as dimension-less two-dimensional scalar fields, whereby the local dynamics ofu is much faster than that of v (ε≪1). The Laplacian D∇2u i, j, Deffectively being the diffusion coefficient, is integrated into thenumerical scheme via a first-order numerical approximation D(u i −1, j+ u i +1, j+ u i, j− 1+ u i, j + 1−4u i, j), so that the connectionbetween nearest-neighbour units is established. For parametervalues a=1.05, b=0.01 and ε=0.05, each FitzHugh–Nagumoneuron is described by a single excitable steady stateu = v=0.0. Small perturbations of the excitable steady stateevoke nontrivial spike-like behaviour, which can inducevarious waveforms in the spatial domain of the spatiallyextended system [39]. Thus, without taking into accountinternal stochasticity, which governs the neuronal dynamics,the medium would remain forever quiescent.

To simulate the neuronal dynamics stochastically, we useGillespie's τ-leap method [40], which is an approximation ofthe exact stochastic simulation method [41], but is computa-tionally less expensive. The occurring reaction probabilitiescorrespond to the reaction mechanism governed by thedeterministic Eqs. (3) and (4). In accordance with reactionprobabilities, a discrete change of membrane potential andconductance of potassium channels of the form kx/χ isperformed at each iteration, where kx is proportional to theflux of the corresponding reactant during time τ and χ isthe system size. Such an approach has already been used inthe literature for simulating various biological processes[9–11,14,42,43]. Importantly, χ directly determines the level ofinternal fluctuations to which the medium is exposed. Internal

noise is most noticeable for small system sizes and it vanishes inthe thermodynamic limit given by χ→∞(theoretically), whichinduces deterministic steady state solutions in the dynamics ofeach individual unit and ultimately results in a quiescent medium.

In Fig. 1, characteristic snapshots of the spatial grid for fourdifferent system sizes χ are presented. It can be noticed nicelythat there exists an intermediate system size for which the noise-induced spatial dynamics of themedium ismaximally ordered. Ifχ is small, the excitations are randomly scattered and in case of alarge system size, fluctuations are too small to provoke large-amplitude excitations. Only an optimally pronounced level ofinternal noise is able to induce coherent spatial wavesthroughout the medium. At this point, we emphasize oncemore that the studied spatial dynamics is induced solely byinternal noise. In what follows, we will show that there exists anoptimal system size χ for which a particular spatial frequency ofwaves is resonantly enhanced, thus providing conclusiveevidences for internal noise spatial coherence resonance inexcitable media.

3. Spatial dynamics

To quantify effects of different levels of internal noise on thespatial dynamics of the studied medium we calculate thestructure function according to the equation

P kx; ky! "

¼ hH2 kx; ky! "

i; ð5Þ

where H(kx, ky) is the spatial Fourier transform of the u i,j field ata particular time t and ⟨…⟩ is the ensemble average over differenttemporal realizations of the spatial grid. To study resultsobtained according to Eq. (5) more precisely, we exploit thecircular symmetry of P(kx, ky) as proposed in [30]. In particular,we calculate the circular average of the structure functionaccording to the equation

sðkÞ ¼Z

Xk

P kt# $

dXk ; ð6 Þ

Fig. 2. Circular average of the structure function s(k) for different system sizes χ.

Fig. 3. Internal noise spatial coherence resonance in the studied medium. Signal-to-noise ratio ρ has a clear maximum in dependence on the level of internalstochasticity χ.

212 M. Gosak et al. / Biophysical Chemistry 128 (2007) 210–214


transition in networks of neurons depends on intrinsic noise,which may originate from fluctuations of channel gating,synaptic release, or of background presynaptic activity [34].The role of noise in neuronal networks, and excitable media ingeneral, has been and still is a vibrant avenue of research, thusyielding many experimental and theoretical studies over the pastdecade [35]. While real-life biological neurons exhibit extreme-ly complex behavior, neuronal dynamics must be considerablysimplified in order to make networks computationally traceable.Therefore, the neuronal medium is presently modelled byFitzHugh–Nagumo equations [36,37], which have been derivedfrom the Hodgkin–Huxley model describing the excitabledynamics of electrical signal transmission along neuron axons[38].

Typically, when studying the spatial dynamics of excitablemedia, for example that of neuronal networks, the role ofdifferent intensities of external additive noise has beeninvestigated [32]. Importantly, we presently extend the ideasof previous authors and investigate the spatial order of excitablemedia in dependence on internal noise only, which is introducedby using a stochastic simulation method. We demonstrate that

internal stochasticity is sufficient to extract a characteristicspatial frequency of excitable media, which is best expressed foran intermediate level of internal noise. The phenomenon is thusan observation of internal noise spatial coherence resonance inexcitable media.

2. Mathematical model

As already mentioned above, local units of the excitablemedium under study are governed by the FitzHugh–Nagumoequations [36,37]

dudt

¼ f u ; vð Þ ¼ 1eu 1$ uð Þ u $ vþ b

a

! "; ð1Þ

dvdt

¼ g u ; vð Þ ¼ u $ v: ð2Þ

Individual units are arranged on the L×L (i, j ∈ [1, L])square lattice with no-flux boundary conditions, whereby the

Fig. 1. Characteristic snapshots of the spatial profile of u i,j for χ=(80; 180; 220; 240) increasing from the top left towards the bottom right panel. All snapshots aredepicted on square grids of linear size L=128. The colour mapping is linear, white depicting 0.0 and black 1.0 values of u i,j. For χ=240 the colour scale extends from0.0 to 0.01 in order to reveal small-amplitude fluctuations around the excitable steady state of each unit. Evidently, only for χ=180 the spatial domain is characterizedby periodic excitatory waves, whereas for χ=80 the excitations are randomly scattered. For χ=220 the level of internal stochasticity is too low to induce periodicwaves; only isolated and uncorrelated waves can be noticed, which disappear completely if χ is increased further.

211M. Gosak et al. / Biophysical Chemistry 128 (2007) 210–214

Introductory biophysics A. Y. 2017-18 · Reaction-diffusion processes and morphogenesis Edoardo...

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