The geometry of bacteria colonies I - ELTE
Transcript of The geometry of bacteria colonies I - ELTE
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The geometry of bacteria colonies
The microbiological background of motion, morphology diagram, Fisher
equation. Self-affine surfaces, branching morphology and models for individual
bacteria.
2018
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Heuristic methods in physicsβTraditionalβ approach
β’ You have a well defined question: how is acceleration (a) related to the mass (m) / How is pressure (p) related to the volume (V) in gas?
β’ You fix all the other parameters (keep the external conditions unchanged / molecule number, temperature, etc.)
except those that are of interest:
β You modulate one of them (e.g. m/V)
β You measure the other (a/p)
β Then you plot the results (and you have your answer)
βHeuristicβ approach
β’ You have an (often) less well-defined question about (often) more complex systems: How do bacteria colonies grow? How do fish or birds move?
β’ You assume which might be the relevant parameters (bacteria: nutriment, surface / fish: velocity, density)
β’ You assume a relation between them (creating a model)
β’ Study the behavior of the model (computational models!)
β’ Comparing (some aspects) with the original system
β’ But it still has:β descriptiveβ predictive power
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βHeuristicsβ: is a speculative, non-rigorous argument, that relies on an analogy or on intuition, that allows one to achieve a result or approximation to be checked later with more rigor; (Wikipedia)
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Why not the traditional way?β’ These βmore complexβ systems are usually biological (even social)
systemsβ’ These systems are affected byβ¦
β Temperatureβ Humidityβ Nutrients concentrationβ Group sizeβ Group composition (age, gender, etc.)β The distribution of theseβ Inner social structure of the groupβ Transient inner states of the members (hunger, fear, etc.)β Reproduction stage of the membersβ Rain / sunshineβ A boy throwing a ball into the flock (of birds)β The neighbor cat roaming aroundβ A child crying for ice-cream nearbyβ Etcβ¦
β’ It is hard to create βunchangedβ conditions for measurements3
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Why exactly bacteria colonies?
From a quantitative point of view
β It is possible to βkeep the environment unchangedβ
β A system with interactions that are simple enough to be captured by quantitative models
(The interaction rules are more or less understood)
β A system whose collective behavior can be explored with computational models
(Theories can be modeled and tested via computer simulations)
β They can give an insight into the formation of self-organized biological structures
Colony of Paenibacillus vortex bacteria
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Why exactly bacteria colonies?From a biological point of viewβ¦
β Unicellular organisms
β Living in colonies
β They are easy to handle in experiments
β They produce various spatio-temporal patterns
β The patterns are often independent of the interaction details - βuniversalityβ
β Dependency on environmental conditions
β Experiments can be reproduced
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The set-up of the simplest experiments for colony formation
Bacteria are grown on the surface of agar gel (an alga)
β βDryβ surface ( = big agar concentration)
β’ The cells can not move (to spread over the substrate can take even weeks)
β’ The duplication time is much smaller β proliferation is the key factor in determining the morphology
β βSoftβ gel (= small agar conc.)
Or: the bacteria produce surfactant
β’ The colony spreads over the substrate in a few hoursβ bacterial motion and chemotaxis are the main factors 6
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Microbiological background - Proliferation
β’ Growth (the increase of the number and total mass of bacteria) strongly depends on the nutrient concentration
β’ Rate of growth (number of cell divisions within a population of unit size during a unit time interval) increases with the nutrient concentration in a hyperbolic manner.
A certain amount of nutrient is required to
maintain the intracellular
biochemical processes
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Procaryotes move in aquatic environment by rotating their flagella
Bacteria can haveβ’ One flagellum, βmonotrichousβ
β’ A pair of flagella at the opposite cell poles, βamphitrichousβ
β’ Clusters of flagella at the poles, βlophotrichousβ
β’ Uniformly distributed flagella over the cell membrane, βperitrichousβ
Microbiological background - Motility
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The direction of flagellar rotation determines the motion
The forward motion is interrupted by short intervals of βtumblingβ
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β’ Entirely different type of motility (flagella-independent)
β’ Slower and smoother than swimming
β’ Requires surface contact
β’ Employed by many strains when moving on surfaces
β’ No visible cellular structures associated β little is known about it
β’ Slime secretion
β’ Motion types varies greatly β probably more than one mechanisms existβ Gliding along the direction of the long axis of the cell (e.g.
Myxococcus or Flexibacter)
β Screw-like motion (e.g. Saprospira)
β Direction perpendicular to the long axis (Simonsiella)
Bacterial Motility - Gliding
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β’ Bacteria are attracted by nutrients (sugar, amino acids, etc.)and repelled by harmful substances and metabolic waste products.
β’ Other environmental factors, e.g. temperature, light, oxygen concentration
Microbiological background - Chemotaxis
β’ Stochastic process: chemical gradients modulate the tumbling frequency: repressed when moving towards chemoattractants
β’ A molecular machinery compares the changes of the chemical concentration in time.
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Random and biased walks. Left: A random walk in isotropic environments. When the cell's motors rotate CCW, the flagellar filaments form a trailing bundle that pushes the cell forward. When one or more of the flagellarmotors reverses to CW rotation, that filament undergoes a shape change (owing to the torque reversal) that disrupts the bundle. Until all motors once again turn in the CCW direction, the filaments act independently to push and pull the cell in a chaotic tumbling motion. Tumbling episodes enable the cell to try new, randomly-determined swimming directions. Right A biased walk in a chemo-effector gradient. Sensory information suppresses tumbling whenever the cell happens to head in a favorable direction. The cells cannot head directly up-gradient because they are frequently knocked off course by Brownian motion.
Source: http://chemotaxis.biology.utah.edu/Parkinson_Lab/projects/ecolichemotaxis/ecolichemotaxis.html
Microbiological background - Chemotaxis
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https://www.youtube.com/watch?v=zlFRJftA2bU
Microbiological background: Bacterial Motility. 4:35 mins
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Bacterial Flagellum. 5:30 mins
https://www.youtube.com/watch?v=v1NnMmw8v80 14
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Morphology diagram
β’ A diagram showing the shape (morphology ) of the bacterium colony as a function of certain environmental parameters β temperature, humidity, chemical composition of the substrate, etc.β Can result in different morphologies even for the same strain
β’ Characteristic colony shapes are assigned to the parameter pairs
β’ Most systematic experiments explore the relation between the concentration of the agar and nutrients.
β’ Agar concentration (consistency of the gel) determines:β’ motility of the bacteria and β’ diffusibility of the nutrient
β’ Nutrient concentration determines:β’ the proliferation rate
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Morphology diagram of Bacillus Subtilis
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Morphology diagram of Paenibacillusdendritiformis
βAβ: Fractal
βBβ: Compact with rough boundary
βEβ: Dense branching
βFβ: On hard substrate a new, βtwistedβ morphology appears
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βSummaryβ of the morphology diagrams
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Compact morphology
Abundant nutrient β compact colony
Either smooth or irregular perimeter
Soft gel dry gel
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Compact morphology
Abundant nutrient β compact colony
Either smooth or irregular perimeter
Soft gel β - Bacteria can move
- Takes a few hours to migrate across the dish
- Random walk trajectory
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Compact morphology
Abundant nutrient β compact colony
Either smooth or irregular perimeter
Soft gel β - Bacteria can move
- Takes a few hours to migrate across the dish
- Random walk trajectory
Inter-cellular interactions are negligible
Time dependence of the bacterial density Ο can be described by the Fisher-Kolmogorov equation
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Fisher-Kolmogorov equation
Starts as a small spotβ Diffuses due to random translation, and multiplicates
ππ
ππ‘=π·ππ»
2π + π(π, π)
Notations:
π = π π, π‘ : bacterial density
π·π : Diffusion coefficient (can be measured as the average
displacement of the cells within a given time interval β see later)
π» : Partial derivative with respect of the space coordinates
π = π(π, π) : Bacterial multiplication
c : Nutrient concentration22
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Fisher-Kolmogorov equation β cont.ππ
ππ‘=π·ππ»
2π + π(π, π)
β’ DΟ, (diffusion coefficient) can be determined from the (measurable) squared displacements d2(t) of the individual cells during a time period t as :
π2 π‘ = 2π·ππ‘
(overline: averaging among the cells)
β’ f(Ο,c): bacterial multiplicationβ When Ο is small, cells proliferate with a fixed rate
β exponential growthβ In practice, even with unlimited nutrient supply, thereβs a
certain threshold Ο* for the density (e.g., accumulation of toxic metabolites)
β We choose cell density units such that Ο* =1 23
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Fisher-Kolmogorov equation β cont.ππ
ππ‘=π·ππ»
2π + π π, π
β We choose Ο*=1 (threshold-density, above which cell-density can not increase)
β Growth rate decreases as Οβ Ο*=1, and f(1)=0
β The specific form of f is unimportant, be will use
π π, π = π π β π(1 β π)
which satisfies the above criteria
β R (c) is a function expressing
how the proliferation depends on the
nutrient concentration
R(c)= for small c values R~c
for big c values R is constant 24
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Fisher-Kolmogorov equation β cont.
Dependency on Ο :
When Ο is small, cells proliferate with a fixed rate
β exponential growth
In practice, even with unlimited nutrient supply, thereβs a certain threshold Ο* for the density (e.g., accumulation of toxic metabolites)
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R(c)= for small c values R~cfor big c values R is const.
Some amount is needed for maintaining the intracellular biochemical process
π π, π = π π β π(1 β π)
Dependency on c :Hyperbolic manner
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Numerical solution of the Fisher-Kolmogorov equation in 1 D
ππ
ππ‘= π·ππ»
2π + π π, π = π·ππ»2π + π (π)π(1 β π)
Numerical solution: the growing domain of the colony expands with a constant speed π£ β π£β where
π£β = 2 π π·π26
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Compact morphology
Abundant nutrient β compact colony
Either smooth or irregular perimeter
β’ Dry gel and/or un-motile bacteria
β Bacteria exert mechanical pressure on their environment (in order to expand to their preferred size)
β Inter-cellular interactions
β Modified Fisher-Kolmogorov equation
β Irregular (self-affine) surface
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Cell-cell interactionβ’ When the bacteria are not independent during the spreading of
the colony (e.g. non-motile cells)β Abrupt change in the cell density at the border of the colony
β’ Propagation of the boundary: expansions of the cell volumes inside the colonyβ The bacteria can not expand to their preferred size, they exert
mechanical pressure on their surroundings
β Large densities: p ~ Ο-Ο0 (Ο0 threshold density for close-packed colonies)
β’ For large density values the displacement is: π£ = π·0π»(π β π0)(D0: diffusion coef., similarly to DΟ in the F-K. eq.)
β’ Modified F-K. eq: ππ‘π = π·0π»
2π + π π for π > π0π π otherwise
β’ In such cases the colony boundary is self -affine 31
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The boundary of a Bacillus subtilis colony (OG-01 strain) grown on hard agar
Long bundles of chains of individual cells consist the colony of the B. subtilis OG-01 strain grown on hard agar. Note the abrupt
change in cell density at the boundary.32
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The formation of self-affine boundaries β the Eden model
β’ One of the earliest method to generate self-affine objects (1961)
β’ Cells grown on a lattice
β’ One single rule for growing the colony:
β In each step, one of the lattice sites
next to the populated areas is
chosen randomly and occupied.
- Or: in each time step, a randomly
chosen (non-motile) bacterium proliferates.
β’ Primitive, but universal model33
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Eden-model
β’ Initial step:
1 occupied cell
β’ Variants:β Each position with same probability
β Higher number of occupied neighbors increase the probability
β’ Variants of the model leave the statistical features of the developing clusters invariant in the asymptotic limit.
A typical colony in the Eden model grown on a strip of 256 lattice units.
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Simulations of the Eden model in 2D
β The lattice can destroy the
rotational symmetry
β Continual model is more
realistic
https://youtu.be/hluvLTwMFOs
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Summary of the Eden model
β’ The surface contains βoverhangsβ
β’ Basic assumptions:β The units can not move (no βdiffusionβ)
β Multiplication on the surface
β’ The model is simple but can be applied to many phenomena β βuniversalityβ
β’ The result is a self-affine surface
β’ KPZ modelβ The time evolution of the profile of a growing interface
β Kardar, Parisi, Zhang: Dynamic scaling of growing surfaces. Physical Review Letters (1986)
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The Kardar-Parisi-Zhang (KPZ) equation
ππ‘β = πππ₯2 β +
π
2(ππ₯β)
2+π’ + π
β’ β : Height of the surface
β’ ππ‘ : Partial derivative with respect to time
β’ ππ₯ : Partial derivative with respect to the space
coordinate x; ( ππ₯2 : second derivative )
β’ π : surface tension coefficient (nu)
β’ π’ : growth speed, perpendicular to the surface
β’ π : uncorrelated noise (stochastic)37
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The KPZ step-by-step
Speed of vertical growth:
ππ‘β π₯, π‘
Components:
1. Surface tension term πππ₯2 β
β 2nd derivative negative β local max (βtop of a humpβ)
β 2nd derivative positive β local min (βbottom of a swaleβ)
β Tends to smoothen the interface
β Does not permit discontinuities (large jumps) in h
β π: surface tension coefficient
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ππ‘β = πππ₯2 β +
π
2(ππ₯β)
2+π’ + π
β’ β : Height of the surface
β’ ππ‘ : Partial derivative with respect to time
β’ ππ₯ : Partial derivative with respect to the space
coordinate x; ( ππ₯2 : second derivative )
β’ π : surface tension coefficient (nu)
β’ π’ : growth speed, perpendicular to the surface
β’ π : uncorrelated noise (stochastic)
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The KPZ step-by-step
Speed of vertical growth: ππ‘β π₯, π‘2nd component: makes the surface lumpy
ββ =π’ β βπ‘
cosπ= π’ β βπ‘
1
cosπ= π’ β βπ‘ 1 + π‘π2π β
β π’ β βπ‘ 1 +π‘π2π
2= π’ β βπ‘+
π’ββπ‘
2π‘π2π β
β π’ β βπ‘+π’ β βπ‘
2(ππ₯β)
2
During a small Ξt period of time the growth of the surface:ββ
βπ‘β π’ +
π’
2(ππ₯β)
2
Due to other effects π’
2β
π
2(more general equation)
1Dβ2D
β’ π₯ β r 1 + π‘π2π₯ =1
πππ 2π₯;
β’ ππ₯ β π» if Ι<<1, then 1 + ν β 1 +π
2if π βͺ 1, then tg (π) β ππ₯β
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ππ‘β = πππ₯2 β +
π
2(ππ₯β)
2+π’ + π
β’ β : Height of the surface
β’ ππ‘ : Partial derivative with respect to time
β’ ππ₯ : Partial derivative with respect to the space
coordinate x; ( ππ₯2 : second derivative )
β’ π : surface tension coefficient (nu)
β’ π’ : growth speed, perpendicular to the surface
β’ π : uncorrelated noise (stochastic)
cosπ =π’ β βπ‘
ββ
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The KPZ step-by-step1Dβ2D
ππ‘β π, π‘ = π β π»2β π, π‘ +π
2π»β
2+ π’ + π π, π‘
β’ Smoothening component (surface tension)
β’ Roughening
β’ noise: π= π π, π‘ : stochastic (=non-deterministic), uncorrelated in space and time
Comments:β’ In case of uncorrelated π π, π‘ noise the resulting surface is self affine
β’ In this case (and in the Eden model) the roughness exponent H=1/2, in contrast to experiments, where Hβ0.7, β¦, 0.8
β’ Reason: in the KPZ the noise is uncorrelated in time (β reality!) 40
ππ‘β = πππ₯2 β +
π
2(ππ₯β)
2+π’ + π
β’ β : Height of the surface
β’ ππ‘ : Partial derivative with respect to time
β’ ππ₯ : Partial derivative with respect to the space
coordinate x; ( ππ₯2 : second derivative )
β’ π : surface tension coefficient (nu)
β’ π’ : growth speed, perpendicular to the surface
β’ π : uncorrelated noise (stochastic)
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KPZ with quenched noise
β’ Uncorrelated noise (in time):
β If the noise is π π, π‘ at the position π at time π‘, then the noise is βindependentβ of π π, π‘ at the same place, at time π‘ + βπ‘.
β’ In other words:
β If the spreading of the colony sticks at time π‘ at position π due to the local inhomogeneity π π, π‘ of the surface (gel), then at the same position, Ξt later, the noise would be independent (uncorrelated), that is, the surface would move on.
β’ In contrast, the reality is that
β Such noises are often constant in time
β The colony moves in an inhomogeneous medium, in which the inhomogeneity is constant in time
β The noise βquenchesβ into the medium. βquenched noiseβ 41
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β’ If the noise is constant (and fixed) in time:
β If the spread of the colony surface sticks at a given point π, then this βhaltβ can be extensive in time, since the media does not change.
β Results in a surface proceeding in a hoping/jiggling manner (points are blocks).
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KPZ with quenched noise
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KPZ with quenched noise
β’ Defining the π π, π‘ quenched noise:
β Let us consider a Ξ(u) function with the following properties:
β’ If u is close to 0, then β π’ β 1 (in a small, finite interval), and
β’ Everywhere else Ξ(u)=0.
β’ a βblurredβ Dirac-delta
β π π, π‘ β 2π· π( π, β( π, π‘))
β’ π is normalized noise
β’ whose spatial autocorrelation is πΆ π π, πβ² = Ξ( π )Ξ( πβ² )
β That is, correlated in a very small spatial interval
β’ D : average magnitude of the noise as πΆ π(0,0) = 2π·
β We incorporate this quenched noise into the KPZ, we get:
ππ‘β π, π‘ = π β π»2β π, π‘ +π
2π»β
2+ π’ + π π, β( π, π‘)
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KPZ with quenched noise
ππ‘β π, π‘ = π β π»2β π, π‘ +π
2π»β
2+ π’ + π π, β( π, π‘)
β’ By βappropriateβ choice of the time and length units the parameters Ξ», Ξ½ and u can be transformed out
β the Ξ»=Ξ½=u=1 case:
ππ‘β = π»2β +1
2π»β
2+ 1 + π = π»2β + 1 + π»β
2+ π
(where the magnitude of Ξ· is ππ = πΆ π(0,0) = 2π·)
β’ Two extreme cases:
1. π· βͺ π·β~1
2. π· > π·β~1
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KPZ with small quenched noise
β’ First Case: π· βͺ π·β~1
β’ The interface is never pinned, advances with a steady velocity
β’ Fluctuating noise with some finite temporal correlations
β’ The standard KPZ can be applied
β’ Resulting interfaces with H=1/2.
β’ Experimental support: Colonies grown on soft agar gel (small pinning effect) showed standard KPZ-like behavior with surface characterized by H=1/2.
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KPZ with big quenched noise
β’ Second Case: π· > π·β~1
β’ The interface is pinned at some certain points, for an extended period of time (until the neighboring segments pull it out)
β’ If the density of the pinning points is high enough, then the propagation of the whole surface can be blocked.
β’ The shape of the frozen colony is determined by the distribution of these pinning sites (and independent of the growth dynamics).
The surface roughening can be mapped onto a directed percolation problem:
finding directed and connected paths
Let us consider a lattice instead of the continuous case
(discrete model, regarding both h and the location (x,y) )
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Directed percolation
β’ The chain of the pinning sites define a directed percolation cluster (if it exists).
β Complete blocking of the interface propagation appears when there is a directed, connected path (a directed percolation cluster)
β The propagation stops along these clusters 47
1. Let us define each lattice site as 2. βpinningβ with a probability 0<p<1.
(gray squares)3. We start from one end of the panel4. On the pinning sites we can move
ahead, up and down (but not backwards)
5. Do we reach the other end of the board?
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Correlation lengths of directed percolation clusters
β’ DPC is characterized by two correlation length:1. Parallel to the interface (to the preferred direction) πβ₯2. Perpendicular to the interface (to the preferred direction) πβ₯
β’ There is a critical probability pc (defining the density of the pinning sites)πβ₯~ π β ππ
βπβ₯ and πβ₯~ π β ππβπβ₯
withπβ₯ = 1.733 and πβ₯ = 1.097 (numerical results)
β’ The width of the interface: π€ β πβ₯β’ Complete blocking of the interface when πβ₯=L
(L is the system size)
β’ The width of the interface at the critical point:
πΏπ»~π€ β πβ₯~ π β ππβπβ₯~πβ₯
πβ₯πβ₯ β πΏ
πβ₯πβ₯ β π» =
πβ₯
πβ₯= 0.633
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Directed percolation
β’ The numerical result for the (discrete) directed percolation problem is H=0.633 β (complete blocking)
β’ Experimental results: Hβ0.7-0.8
β’ Reason: the observed colonies have both blocked and freely moving parts β higher roughness exponent (H) than for the blocked interface.β Numerical simulations:
Hβ0.71-0.75
(close to the observations)
β’ KPZ with quenched noise
and the DP simulations
have the same results49
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Branching morphology
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β’ Nutrient-poor agar substrateβ’ Complex, branching coloniesβ’ Not exhibited by all strains
(but by many)
Baillus subtilis colony, under nutrient-poor conditions. 8
and 19 days after inoculation.
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Branching morphology β colony formation
β’ Basic assumption: β the growth of the colony is diffusion-limited:
β The multiplication of the bacteria is determined by the locally available nutrientβ’ At the beginning: local nutrient is enough to maintain the
growth
β’ After some bacterial multiplication, nutrient deprivation progresses in and around the colony
β’ Further growth depends on the diffusive transport from distant regions of the petri-dish
β Experimental supportβ’ Non motile B. subtilis grows
only towards nutrient-rich
regions 51
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Branching morphology β colony formation
β’ The speed of the growth is determined by the nutrient diffusion
β’ The colony develops towards the nutrient
β’ Instability:β Due to some random perturbation a small part of the
colony advances βaheadβ (towards some nutrient)β This part of the colony gets closer to the nutrientβ Can multiply faster
β’ This process stops at a certain curvatureβ Certain amount of neighboring cells are neededβ A certain βsteady shapeβ is set
β’ New perturbation: new branch 52
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Diffusion-Limited Aggregation (DLA)
β’ The definition of the basic DLA algorithm:
β Start: 1 cell
β In each time step:
β’ A particle (performing random walk) departs from infinity
(in simulations from finite distance)
β’ Sticks to the colony upon graze
β’ Result: Fractal-type clusters
53Typical DLA cluster with 50, 000 particles
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Relation to bacterium colonies
β’ Random walk of the particle ~ diffusion of the nutrient
β’ Sticking to the colony ~ bacterium proliferation
β’ Non-motile bacteria!
β’ Very simple model (1 βnutrient-unitβ = 1 multiplication) generating realistic formations β βuniversalityβ
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Refinement of the DLA model βModeling non-motile bacteria
β’ Assumptions:β Bacteria interact with each otherβ Each particle (cell) is characterized by
β’ Space coordinate xi
β’ Energy state Ei (or cell cycle state)β Ei <0 : spore state. Without nutrient, remains in this stateβ 0<Ei <1 : right after multiplication β Ei >1 : has enough energy to multiply
β’ Notations: ππ : nutrient consumption rate π : conversion factor relating the maximal nutrient consumption rate with the
shortest cell cycle time (nutrient β energy conversion)
π : generic βmaintenanceβ term (not directly contributing to growth)
The energy-level of bacterium i: uptake - consumption
ππΈπππ‘
= π β ππ β π56
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Modeling non-motile bacteria βlimits of the nutrient uptake
β’ Further notations: ππππ₯ : maximal nutrient uptake rate of the cells π : efficiency of the enzymatic reaction converting the nutrient
into energy π(π₯π) : nutrient concentration (around cell i) π(π₯π) : local cell density π0π : maximal diffusive transport from the substrate to the cell ππ : nutrient consumption rate (of bacterium i)
β’ The rate with which the cell-mass grows:
π π₯π ππ = πππ ππππ₯π π₯π , π0π π₯π
β The nutrient-uptake is limited by the enzymatic rates and local nutrient concentration(the maximal speed with which cell i can take in the nutrient)
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Modeling non-motile bacteria βHow the local nutrient concentration varies
β’ Bacteria use up the nutrient
β’ Changes in c are given by the diffusion equation with the appropriate sink terms at the position of the active particles:
ππ
ππ‘= π·ππ»
2π β
π
πππΏ(π₯ β π₯π)
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Diffusion Sinks: the cell at location xi consumes the nutrient with rate Οi
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Summary: non-motile bacteria in nutrient-poor environment
(i) The energy-level of cell i
ππΈπππ‘
= π β ππ β π
(ii) Cell-mass growth rate
π π₯π ππ = πππ ππππ₯π π₯π , π0π π₯π
(iii) Changes of the local nutrient concentration
ππ
ππ‘= π·ππ»
2π β
π
πππΏ(π₯ β π₯π)59
πΈπ : energy level of cell iπ : efficiency of the enzymatic reaction converting the nutrient into energyππ : nutrient consumption rateπ : generic βmaintenanceβ term (not directly contributing to growth)π(π₯π) : local cell density ππππ₯ : maximal nutrient uptake rate of the cellsπ(π₯π) : nutrient concentration (around cell i)π0π : maximal diffusive transport from the substrate to the cellπ·π : nutrient diffusivity
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Results: Modeling non-motile bacteria with the refined DLA model
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Simulation results: Morphology diagram generated by the model with non-motile particles as a function of the initial nutrient concentration (c0)and nutrient diffusivity (Dc). The
colonies were grown (in the computer) until either their size or the number of bacteriareached a threshold value.
Related experiments: the morphology diagram of
the non-motile B. subtilisOG-01b strain
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Motile bacteria with the DLA model
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β’ Rules (i)-(iii) remain the sameβ (i) energy-level of cell i β (ii) cell-mass growth rateβ (iii) changes of the local nutrient concentration
β’ Non-motile cells β self-affine surfaceβ’ Motile bacteria β smooth surface. But they still can not migrate
on dry agar surface, only where some surfactant was secreted:β’ New rules:
β (iv) The active particles move randomly (with Brownian motion) within a boundary:
ππ₯πππ‘
= π£π π
where π is a unit vector pointing in a random directionβ (v) The propagation of the bacteria is assumed to be proportional to
the local density of the active cells. Collisions of the particles with the boundary is counted, and when a threshold value (Nc) is reached, the neighboring cell is occupied as well. (the boundary shifts forward)
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Results: Modeling motile bacteria with the refined DLA model (on hard agar gel)
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Morphology diagram generated by the model with motile bacteria as the function of the initial nutrient
concentration (Co) and agar gel βhardnessβ, (the threshold value Nc for the colony borderline displacement).
Corresponding experimental results: Morphology diagram
of Paenibacillus dendritiformis. Agreement with the model
results within a limited region of the parameters, but it fails
to predict the formation of the thin, straight radial branches
at very low food concentrations.
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Simulation results:
β’ Nice agreement with the experiments, butβ’ Fails to explain the transition between the
Fractal-type and non-fractal-type colonies
Solution: assuming repulsive chemotaxis signaling among the cells. Due to the repulsion the cells by-pass each other: the random Brownian motion becomes biased.
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Simulation results without (a) and with (b) repulsive chemotaxis signaling.