Stas_lecture2

Stas_lecture2 - From Microscopic Parameters to Macroscopic...

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From Microscopic Parameters to Macroscopic Balances (Expression for the Chemotactic Flux) Random Motility and Chemotaxis Macroscopic Flux (1) () n vn n n tx n vn n n λλ + +− + + −+ + ∂∂ += −= total cell density: flux: ( ) nn n j vn n ≡+ ≡− 2 steady state : eq nv vn v v n xx j λ −− − = +
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Macroscopic Flux (2) 1 2 [ ] persistence time random motility coefficient ( ) chemotactic velocity p p cp T Tv VT v λλ µ −+ ≡+ ≡− eq c p nv jV n T v n x x ∂∂ =− + Three contributions to flux: 1. random motility 2. chemotaxis (right- and left- moving cells reverse differently) 3. chemokinesis (gradient in cell velocity) in phenomenological models eq n jn x µα + To couple to external concentration field, combine with the experimentally determined dependencies of and T p Flux in a 1D Gradient (1) Motivated by Berg & Brown 1972 Experiments • runs & tumbles • tumble duration is zero • use velocity jump process in 1D • motion in a gradient 1 // () (1 ) / 2 : is the tumbling probability : "directional persistence" probability of reversing after tumbling T p T T p p λψ ψ +− =+ random motility chemotaxis gradient run time B dN C dt receptor-mediated mechanism: # of occuppied receptors B N Flux in a 1D Gradient (2) time derivative seen by the “bacterium” 0 b dN dt e σ ττ = relate to the frequency of tumbles 0 1/ B dN dt T pp e τ == B BB dN N N v dt t x 0 )cosh( ) )sinh( ) b b N v x N pv x +=− −= − 2 0 sec ( ) ) tanh( ) b b c N v hv px N Vv v x µσ = −∂ =
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Flux in a 1D Gradient (3) Simple Ligand/receptor Equilibrium 2 () total T d B B Dd Nc N K dN N K cd cK c = = ++ 2 1 0 [cosh( )] (1 ) tanh( ) B B c vc d N v px d c cdN Vv v x dc µσ ψ σ = −∂ = 2 2 0 , ) B c vd N c pd c x == small gradients: If the model is correct: macroscopic flux can be estimated from data on binding and microscopic parameters for cell migration chemotactic coefficient, χ Flux in a 1D Gradient (4): Analysis C x c V C x µ 1. Random motility coefficient increases with temporal gradient 2. Random motility coefficient is a decreasing function of spatial gradient: at large gradients all cells swim in one direction 3. Chemotactic velocity has a limiting value: the population can not move faster than the maximal cell speed Cell density + diffusing signal
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“Chemotactic Wave Paradox” Observation aggregation to the source of chemical wave pulse of cAMP is nearly symmetric Devreotes & Tomchik, Science 212, 443-6, 1981 Simple-model: symmetric chemotactic velocity no net directed motion
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Stas_lecture2 - From Microscopic Parameters to Macroscopic...

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