Fundamentals of Bio-architectureSUMMARY

Melik Demirel, PhD

*Pictures and tables in this lecture notes are copied

from Internet sources for educational use only.

Can order increase without breaking 2nd law of Thermodynamics?

1. Reaching equilibrium (Prigogine)

2. Oscillatory reactions out of equilibrium (Instabilities)

Max-Planck-Institut für Metallforschung, Stuttgart Abteilung Neue Materialien und Biosysteme

Diblock Copolymer Self-Assembly

Frank Bates et al.

http://www.cems.umn.edu/about/ people/faculty.php?id=20164

Tom Russell et al. http://www.pse.umass.edu/trussell/

Diblock Copolymer Assembly

1. REACHING EQUILIBRIUMSystem dissipate energy as heat, and system gained entropy, equilibrium structure

obtained (surrounding entropy increased!)

Phase diagram of block copolymers

Non-equilibrium Pattern Formation: Spinal decomposition

2 Coarsening rates in binary viscous fluids

authors obtain the same lower bound on the energy as in the cont inuum case.

In [23], the authors consider a more general class of non-convex funct ionals

classified by an exponent α for largem-values. They find opt imal lower bounds

on the energy with an exponent which depends on α.

In our analysis, we adopt the method from [36] to a model that allows for diffusive

and convect ive transport .

2.3 T he model

We are interested in the late-stage coarsening phenomenon which occurs when a

thermodynamically unstable two-phase system, in our case a binary viscous fluid

quenched slight ly below a crit ical temperature, demixes, i.e., the two phases sepa-

rate into two domains of the two different equilibrium volume fract ions. As order

parameter, which locally describes the composit ion of the mixture at any one t ime,

we consider a scalar field

m(t, x) ∈ .

The free energy is given by the Ginzburg–Landau funct ional

E(m) = −1

2|∇ m|2 +

1

2(1− m2)2 dx, (2.1)

which favors locally the separat ion of the system into its two phases, encoded by the

values − 1 and 1, and penalizes transit ions between domains occupied by different

phases. Hence length, energy, and order parameter have been nondimensionalized

such that the equilibrium values, the width of the interfacial layers, and the en-

ergy per area of the interfacial layers are all of order one. Furthermore, we have

normalized the free energy by the volume of the system, denot ing the average by

− dx.

The evolut ion must conserve the volume fract ion of each phase separately. We are

interested in the case of a “ crit ical mixture” , where both phases occupy the same

volume fract ion, i.e.,

m = − m dx = 0. (2.2)

This restrict ion is rather a notat ional convenience than a mathemat ical necessity.

In fact, our results can be easily generalized to any volume fract ion. Not ice that

in the case of off-crit ical mixtures, i.e., 0 < m 1, the growth rate for diffusion

mediated coarsening shows an explicit dependence on m, cf. [11].

The evolut ion equat ion for m takes the form of a conservat ion law:

∂tm + ∇ · J = 0,

for some flux J . The flux takes account of the two parallel t ransport mechanisms:

diffusion and convect ion. Diffusion is the relat ive mot ion of the unlike part icles.

10

m: composition of mixing

2. Oscillatory reactions out of equilibrium

Does it violate 2nd law of thermodynamics: (Ofcourse!) No

belousov-zhabotinsky (BZ) reaction

Belousov, a Russian Biochemist, were unable to publish his results initially!!!

Later Zhabotinsky (as a graduate student) rediscovered Belousov’sfindings

Complex Patterns

Leiseggang Patterns

Brain imagingBZ reaction

Stripes in animal kingdom

Why do animals’ coats have patterns like spots, or stripes?

Biological symmetry

Common Snail (Helix)Ovulate Cone (Pinus)Muscadine Grape Tendril (Vitis rotundifolia)

A type of symmetry in which an organism can be divided into 2 mirror

images along a single plane.

A packing arrangement in which the individual units are tightly packed regular hexagons. There is no more efficient use of packing space than this, and it occurred first in nature.

A symmetry based on the pentagon, a plane figure having 5 sides and 5 angles

Spirals Bilateral Symmetry

Hexagonal Packing Pentagonal Symmetry

Turing Instability: Reaction-Diffusion Model

Mullins Sekerka Instability

Molecular Pattern Formationall snowflakes have 6 sides because of

intermolecular forces.

Bacterial Pattern Formation

Cell-Surface Interactions and Pattern Formation

Mechanical Theory of Generation Pattern and Form

Linear Stability Analysis

Swift-Hohenberg Equation

Examples of Symmetry Breaking

• Belousov-Zhabotinsky Reaction (BZR)

• AngleFish Stripe formation

2. Oscillatory reactions out of equilibrium

Does it violate 2nd law of thermodynamics: (Ofcourse!) No

belousov-zhabotinsky (BZ) reaction

Belousov, a Russian Biochemist, were unable to publish his results initially!!!

Later Zhabotinsky (as a graduate student) rediscovered Belousov’sfindings

2. Oscillatory reactions out of equilibrium

• Alfred J. Lotka

(1880-1949)

– American mathematical biologist

– primary example: plant population/herbivorous animal dependent on that plant for food

• Vito Volterra

(1860-1940)

– famous Italian mathematician

– Retired from pure mathematics in 1920

– Son-in-law: D’Ancona

A system reaching steady supply of energy (oscillatory reactions) can reach steady state

Lotka-Volterra Model

xyyy

yxx

Introduction to Ordinary Differential EquationsStephen Sapesrtone

x: Prey or Activator

y: Predator or Inhibitor

Reaction-Diffusion Model

Pattern Formation

• Patterns can be …– Time dependent (periodic in time or space)– Transient or persistent– Free energy away from equilibrium to maintain

pattern (thermo – dissipative structure)

• Turing Theory and Pattern Formation– Steady state stable to homogeneous perturbations– Unstable to inhomogeneous perturbations– Final structure stationary in time, periodic in space– Intrinsic wavelength– Inhibition diffuses faster than activation

Alan Turing (1952 Phil. Trans. Roy. Soc. )“The Chemical Basis of Morphogenesis”

Diffusion

• u(t,x) : density function of a chemical

• The chemical will move from high density places to lower density places, this is called diffusion

• Diffusion is the mechanism of many molecular or cellular motions

• Diffusion can be described by a heat equation

Reaction-diffusion equations

• Let U(x,t) and V(x,t) be the density functions of two chemicals or species which interact or react

Morphogenesis (from the Greek morphê shape and genesis creation) is one of three fundamental aspects of developmental biology along with the control of cell growth and cellular differentiation. Morphogenesis is concerned with the shapes of tissues, organs and entire organisms and the positions of the various specialized cell types.

• The kinetics are always chosen such that, in the absence of diffusion, the homogeneous steady state is stable (and thus the instability is diffusion driven)

Why do animals’ coats have patterns like spots, or stripes?

Murray’s theory

Murray suggests that a single mechanism could be responsible for generating all of the common patterns observed. This mechanism is based on a reaction-diffusion system of the morphogen prepatterns, and the subsequent differentiation of the cells to produce melanin simply reflects the spatial patterns of morphogen concentration.

Melanin: pigment that affects skin, eye, and hair color in humans and other mammals.

Morphogen: Any of various chemicals in embryonic tissue that influence the movement and organization of cells during morphogenesis by forming a concentration gradient.

“Theorem 1”: Snakes always have striped (ring) patterns, but not spotted patterns.

Turing-Murray Theory: snake is the example of b/a is large.

Snake pictures (stripe patterns)“Theorem 1”: Snakes always have striped (ring) patterns, but not spotted patterns. Turing-Murray Theory: snake is the example of b/a is large.

“Theorem 2”: There is no animal with striped body and spotted tail, but there is animal with spotted body andstriped tail.

Turing-Murray theory: The body is always wider than the tail. The same reaction-diffusion mechanism should be responsible for the patterns on both body and tail. Then if the body is striped, and the parameters are similar for tail and body, then the tail must also be striped since the narrower geometry is easier to produce strips.

Examples: zebra, tiger (striped body and tail), leopard (spotted body and tail), genet, cheetah (spotted body and striped tail)

Spotted body and striped tail or legs

Cheetah (upper), Okapi (lower) Tiger (upper), Leopard (lower)

Spotted body and striped tail

Genet (left), Giraffe (right)

Natural Patterns of cos(kx)

cos(x): Valais goat

(single color: f(x)=1, a lot of examples)

Cos(2x): Galloway belted Cow

cos(2x): Giant Panda