Fundamentals of Bio-architecture SUMMARY

Transcription

1 Fundamentals of Bio-architecture SUMMARY Melik Demirel, PhD *Pictures and tables in this lecture notes are copied from Internet sources for educational use only.

2 Can order increase without breaking 2 nd law of Thermodynamics? 1. Reaching equilibrium (Prigogine) 2. Oscillatory reactions out of equilibrium (Instabilities)

3 1. REACHING EQUILIBRIUM Diblock Copolymer Self-Assembly System dissipate energy as heat, and system gained entropy, equilibrium structure obtained (surrounding entropy increased!) Phase diagram of block copolymers Diblock Copolymer Assembly Frank Bates et al.

4 In [23], the authors consider a more general class of non-convex funct classified by an exponent α for largem-values. They find optimal lower b Non-equilibrium the energy with an exponent Pattern which depends Formation: on α. Spinal decomposition In our analysis, we adopt the method from [36] to a model that allows for dif and convective transport. 2.3 T he model We are interested in the late-stage coarsening phenomenon which occurs w thermodynamically unst able two-phase system, in our case a binary viscous quenched slightly below a critical temperature, demixes, i.e., the two phases rate int o two domains of the two different equilibrium volume fract ions. As parameter, which locally describes the composition of the mixture at any one we consider a scalar field m(t, x). The free energy is given by t he Ginzburg Landau functional E(m) = 1 2 m (1 m2 ) 2 dx, which favors locally the separation of the system into itstwo phases, encoded b m: composition of mixing values 1 and 1, and penalizes transitions between domains occupied by dif

5 2. Oscillatory reactions out of equilibrium belousov-zhabotinsky (BZ) reaction Belousov, a Russian Biochemist, were unable to publish his results initially!!! Later Zhabotinsky (as a graduate student) rediscovered Belousov s findings Does it violate 2 nd law of thermodynamics: (Ofcourse!) No

6 Complex Patterns Stripes in animal kingdom Leiseggang Patterns BZ reaction Brain imaging

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

8 Biological symmetry Spirals Common Snail (Helix) Ovulate Cone (Pinus) Muscadine Grape Tendril (Vitis rotundifolia) Bilateral Symmetry A type of symmetry in which an organism can be divided into 2 mirror images along a single plane. Hexagonal Packing 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. Pentagonal Symmetry A symmetry based on the pentagon, a plane figure having 5 sides and 5 angles

9 Turing Instability: Reaction-Diffusion Model

10 Mullins Sekerka Instability

11 Molecular Pattern Formation all snowflakes have 6 sides because of intermolecular forces.

12 Bacterial Pattern Formation

13 Cell-Surface Interactions and Pattern Formation

14 Mechanical Theory of Generation Pattern and Form

15 Linear Stability Analysis Swift-Hohenberg Equation

16 Examples of Symmetry Breaking Belousov-Zhabotinsky Reaction (BZR) AngleFish Stripe formation

17 2. Oscillatory reactions out of equilibrium belousov-zhabotinsky (BZ) reaction Belousov, a Russian Biochemist, were unable to publish his results initially!!! Later Zhabotinsky (as a graduate student) rediscovered Belousov s findings Does it violate 2 nd law of thermodynamics: (Ofcourse!) No

18 2. Oscillatory reactions out of equilibrium Alfred J. Lotka ( ) American mathematical biologist primary example: plant population/herbivor ous animal dependent on that plant for food Vito Volterra ( ) 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

19 Lotka-Volterra Model x x y y y xy x: Prey or Activator y: Predator or Inhibitor Introduction to Ordinary Differential Equations Stephen Sapesrtone

20 Reaction-Diffusion Model

21 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

22 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

23 Reaction-diffusion equations Let U(x,t) and V(x,t) be the density functions of two chemicals or species which interact or react 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) 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.

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

25 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.

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

27 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.

28 Theorem 2 : There is no animal with striped body and spotted tail, but there is animal with spotted body and striped 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)

29 Spotted body and striped tail or legs Cheetah (upper), Okapi (lower) Tiger (upper), Leopard (lower)

30 Spotted body and striped tail Genet (left), Giraffe (right)

31 Natural Patterns of cos(kx) cos(x): Valais goat (single color: f(x)=1, a lot of examples)

32 Cos(2x): Galloway belted Cow

33 cos(2x): Giant Panda