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Physics of Development
Chaitanya Athale
Outline
• Collec7ve morphogenesis-‐ phenomenology and some theory
Pa<erns of Bacteria
Budrene & Berg (1995) Dynamic forma7on of symmetrical pa<erns by chemotac7c bacteria
1mM succinate with E coli Tsr-‐
2mM succinate with E coli Tsr-‐
3mM succinate with E coli Tsr-‐
3mM succinate with E coli serine blind mutant
• Matsuyama & Matsushita (2001) Popula7on morphogenesis in bacteria. Forma 16:307-‐326
• Morikawa (2003) Simula7on study of bacterial colony with mul7plying rods. Forma 18:59-‐65
Bacteria: Free Swimming
Chemotaxis in liquid media Tumble: change direc7on, Run: straight swimming
Bacillus sub7lis with flagella
Matsuyama & Matsushita (2001) Forma
Swarming Behaviour
• Pa<erns in liquid (Budrene & Berg 1991, 1995) • Surface transloca7on referred to as swarming
Bacteria ac7vely translocate by flagellar rota7on (C, D, E) 1-‐3 days
Spread by volume increase of colony (A, B) 2-‐4 weeks
Dense Branching Morphology (DBM)
Taken from area E (mo7le) Reminiscent of a class of theore7cal models
DBM Structure
Two cell types: (a) Non-‐mo7le wall and 7p cells, (b) Mo7le swirling cells inside
Direc7on of branch extension
swirling
Population Morphogenesis by Cooperative Bacteria 311
the tip wall. The outermost tip wall cells do not move by themselves, instead, the innerswirling cells are pushing out little by little these tip wall cells. Thus, in contrast to themoving behavior of individual cells in swirling cluster, the extension of the branch is quiteslow and unidirectional as shown in Fig. 5. Since bacteria have been thought to behaveindependently of each other, special arrangement of differently working cells in the branchpopulation of B. subtilis (division of labor in the bacterial world) was unexpected finding(MATSUYAMA et al., 1993; MATSUYAMA and MATSUSHITA, 1995). It is interesting thatsimilar differential cell arrangement is present in the tissue at the growing ends of themulticellular organism “plant”.
2.3. Physical factors inducing collective behavior of bacteriaIt is curious that bacteria moving on the surface are always getting together. A single
bacterium separated from others on the surface seemed to be immobile. Even in area Dwhere bacterial population seemed to be spreading homogeneously, translocating bacteriain the spreading front were making raft-like clusters (WAKITA et al., 1994). In the worldof bacteria which are so small, effective working forces are quite different from thoseeffective on organisms of the human size. Instead of gravitation, inter-molecular forcesworking on the surfaces of microorganisms are critical. With the size reduction oforganisms, ratio of surface area to volume of the organism will be greater and surface forcesworking on the organism will be more effective. In addition, water which is indispensablefor life has outstandingly strong surface tension. Thus, water around microbes will restrainthe movement of small unicellular organisms on the surface environments (MATSUYAMAet al., 1992; MATSUYAMA and NAKAGAWA, 1996). To overcome such restricting situationin the small scale surface environment, bacteria seem to evolve the translocation mechanismsby arranging collective cell groups. So, in the area E where nutrients are not enough forenergy supply to every bacterial cell, active cells are gathering at the special structural partin the population. Developing this functional and differentiated population, bacteria as awhole (including immobile cells) seem to be spreading efficiently on the nutrient-poormedium surface. In spite of ubiquitous presence of DBM-like pattern in nature, itsgeneration mechanisms have not been investigated by experimental micro-scale analyses.Morphogenic units of DBM-like pattern shown here are living bacteria which are clearlyvisible under an optical microscope. As shown herein, the microscopic video tracing of thedevelopment processes revealed the precise figures leading to the characteristic pattern,
Fig. 5. Schematic illustration of an extending branch. Open box indicates an vigorously swirling bacterial cell.Closed box indicates an inactive cell. Branch extending direction is indicated by an arrow.
inac7ve Growing branch
Chemicals Inducing Surface Spreading
• Surface ac7ve agents (surfactants)
• Serra7a marcescens
S. marcescens N38
N38-‐09 mutant defec7ve in surfactant “serrawe^n W1” produc7on
Concentric Pa<erns Bacillus sub7lis (phase C-‐ high nutrient, medium s7ffness)
Proteus mirabilis on hard agar medium
Proteus colony dynamics shows intermi<ent growth
Collec7ve Behaviour
Low agar concentra7on ‘Rads’ Hard agar form disk like structure
Differen7a7on
Free living vs. collec7ve modes of migra7on
Chemotaxis Minimal medium agar plate Central disk of filter paper with L-‐alanine (2.5 uM) Proteus inocolated at 4 points
Simula7on of Bacterial Rods
rij=distance between central segments of two rods I and j lb=length of rod a=radius of rod
Simulation Study of Bacterial Colony 61
the bacteria divides into two new individuals at lb = 14a. Therefore the number of rodsincreases with the simulation step and will form a bacterial colony. In the colony, there isa direct repulsive interaction between rods. The potential energy between the rods i and jis represented as
u ra r r a
r aij
ij ij
ij( ) = ( ) >( )
∞ ≤( )#$%
&%( )
2 22
112
/
where rij is the distance between central segments of rods i and j. Since u(rij) includes asoftcore term (2a/rij)12, the rods can push and shove each other (WAKITA et al., 2001). Asa result, more active rods push other rods. This biological repulsion corresponds to a “short-range repulsive chemotaxis” (KOZLOVSKY et al., 1999).
The movement of the rod is classified as passive or active (BERG, 1992). A passivemovement is caused by a fluctuation of surrounding mediums and thus it depends on thetemperature T and the viscosity of the mediums. From the Stokes’ law, the mean displacementδP
2 = 2Δτk T fB xy/ and the mean rotational angle θP2 = 2Δτk T fB r/ of the rod
for a unit time Δτ are adopted (BERG, 1992). Here fxy = 3πηlb/ln(lb/a) is the viscous dragcoefficient of the rod moving at random and fr = πηlb
3/3(ln(lb/a) – 1/2) is the rotationalfrictional drag coefficient of the minor axis. kB is Boltzmann’s constant. η is the coefficientof viscosity of the surrounding mediums, which depends on not only the concentration ofagar but also the lubricant such a surfactant secreted by the bacteria. An increase in theamount of lubricant decreases the friction between the bacteria and the agar surface. Froma reaction-diffusion model for a bacterial colony including a time-evolution equation of alubricant, Kozlovsky et al. suggested that the coupling of the bacterial motion to thelubricant should be replaced by a density-dependent diffusion coefficient for the bacteria(KOZLOVSKY et al., 1999). We adopt the suggestion into our model and define thecoefficient of viscosity η = η0/s(n) for each rod i, where n is the number of surrounding rodswhich are less than 10a from the rod i. Since the function s(n) is regarded as an increasingfunction, we assume s(n) = 2n × 0.01 + 0.99.
On the other hand, the active movement of a bacteria is driven by rotation of severalflagellar filaments. When these flagella turn counterclockwise, they form a synchronousbundle that pushes the body steadily forward; this mode is said to “run.” When they turnclockwise, the bundle comes apart and the flagella turn independently. As a result, thebacteria moves in a highly erratic manner; this mode is said to “tumble.” In our model, amimic bacteria also moves due to either “run” or “tumble” mode. In the “run” mode, abacteria goes ahead with the mean displacement δ A
2 = 2ΔτA fb a/ , where the value Ab
represents an “activity” of the bacteria, which equals the amount of nutrients ingested bythe individual. fa is a viscous drag coefficient moving lengthwise written as fa = 2πηlb/(ln(lb/a) – 1/2). In the other mode, “tumble”, a bacteria randomly turns clockwise andcounterclockwise by the mean rotational angle θA
2 = 2ΔτA fb r/ .The alternative modes are determined by a parameter µ which relates to the recent
Poten7al energy Short range repulsive chemotaxis Run Tubmle Viscosity η0 Φij Nutrient triangular la^ce Diffusion and inges7on
Simulated Pa<erns
Swarming in Eukaryo7c Cells
• Tumour growth as a paradigm
Biological Case Studies
1. Tumour growth 2. Collec7ve cell behaviour
Tumour Growth
Glioblastoma mul7forme (GBM)
Washington University, School of Medicine
Why Study GBM Most aggressive and commonest form of human glioblastoma
Death <1 yr of disease Mul.forme: necrosis & haemmorhage,pleiomorphic nuclei & cells, gene7c dele7ons, amplifica7on and point muta7ons, microvascular prolifera7on
Subclones within tumour Tumour of 109 cells might harbour 106 cells with muta7on in any one gene
THERAPY: surgery, chemo-‐ and radio-‐therapy SUCCESS: Long term nil
Holland (2000)
A. Pre surgery B. Post-surgery and radiation-therapy C. Recurrence in 6 months at 2 sites D. Post resection of 2 tumours E. Recurrence of tumour at resection margin
Gene-‐Protein Network in
GBM
Qiagen
Therapeu7c Approaches • Surgical resec7on • Radia7on and chemotherapy
• Mouse and in vitro models • Modified viruses • An7bodies against EGFR, PDGF, Angiogenesis (VEGF) • siRNA • Synthe7c pep7des
Understanding the basis of tumour growth and expansion
Stages of Tumour Growth
Modelling Tumour Growth
• MRI resolu7on ~1mm • In vitro and mouse models do not behave like human
• Single cell events lead to malignant spread
GOAL: Understanding of tumourigenesis at clinically undetectable stage
In Vitro Experiments
Deisboeck et al. (2001)
In Vitro Kine7cs
Emprical Models
Comparison
Experimental volumes (µm3) measured in cell culture tumor spheroids
Discrete Model
Cell as automaton Leaves non-‐diffusive a<ractor trail Path of least resistance carved in medium
Square la^ce (128x128) a= La^ce constant = cell diameter Chemotaxis: Nutrient Homotype a<ac7on: Paracrine
Sander & Deisboeck (2002)
Signalling in a Mul7cellular context
Synthesis
Autocrine
Recycling
Degradation Signalling
Paracrine
Cell 2
Cell 1
Growth Pa<erns
Diffusive growth: Only random walk Compact
χ = chemotaxis coefficient β= drift velocity η=homotype attraction Dc=cellular diffusion coefficient
Chemotac7c and Homotypic growth Disperesed Chain forma7on
Sander & Deisboeck (2002) Growth pa<erns of microscopic brain tumors. Phys. Rev. E 66, 051901 (2002)
Branching Pa<erns
Simula7on results. Very strong homotype a<rac7on and very strong chemotaxis result. (Scale approx. 1 mm).
Experimental Branching
& 7me
Receptor signalling
Synthesis
Autocrine
Recycling
Degradation Signalling
Mul7cellular context
Synthesis
Autocrine
Recycling
Degradation Signalling
Paracrine
Cell 2
Cell 1
Mul7cellular mechanics
Free Surface
Tumor Spheroid
Cells
Division
Migration
NEXT
• Cell adhesion, compartmentaliza7on and forming a hole (lumen)
• Epithelial morphogenesis • Mesenchymal morphogenesis • Pa<ern forma7on: segmenta7on, axes and asymmetry
NEXT • Developmental processes in metazoans • Cleavage and blastula • Cell states • Cell adhesion, compartmentaliza7on and forming a hole (lumen)
• Epithelial morphogenesis • Mesenchymal morphogenesis • Pa<ern forma7on: segmenta7on, axes and asymmetry
• Organogenesis
Scale of Descrip7on
Molecular dynamics: details at atomic and molecular scales
Par7al differen7al equa7ons: pressure, temperature, velocity
La^ce Gases (cellular automata)
Spa
tio-te
mpo
ral s
cale
Complex Systems
Macroscopic systems with many interac7ng microscopic objects
Sophis7cated stochas7c methods used to create simula7ons of complex systems
Capable of exhibi7ng collec7ve behaviour and phase transi7ons
Physical Examples
Freezing or boiling of a liquid Magnet with ferromagne7c or paramagne7c transi7ons
Glass Liquid crystals
Biological Examples Protein folding Membrane organiza7on Spontaneous pore forma7on Targe7ng of transcrip7on factors to genes Cell division Cell migra7on Collec7ve cell migra7on (would healing, development)
Tissue repair and regenera7on Tumours Organ development
nm
µm
mm
ms
µs fs
s
min
hr
La^ce Gas Automata
FHP 2D triangular
la^ce (Frisch, Hasslacher, Pomeau) 1986
x
y
Rules: 1. Collisions Scatter if 2 particles
approach head on Momentum conservation by
random choice 2. Transport
HPP 2D square la^ce
(Hardy, Pomeau, dePazzis)
x
y
FHP Collision Rules
Applicability of LGA for Hydrodynamic Modelling
La>ce Gas Automata Tradi.onal Hydrodynamics
High viscosity inbuilt in rules Flexible viscosity values
Finite spa7al resolu7on No predetermined resolu7on
Imprac7cal for turbulence studies Turbulence can be handled
Complex boundary condi7ons Simple boundary condi7ons
Intui7ve approach Not intui7ve for complex phenomena
Need sta7s7cal averaging Gives a sta7s7cal average
Less flexible for parameter range for generality
Completely flexible and general
Successful applications: flows in porous media, immiscible flows and instabilities, spreading of droplets, wetting phenomena, microemulsion and transport problems Pattern formation, reaction-diffusion, nucleation-aggregation and growth phenomena.
La^ce Gases and Fluids
• Fluid: La^ce gas • Geometry of la^ce • Discrete states permi<ed • Update rule based on neighbours • Compact state descrip7on and lookup table • Conserva7on laws