Modelling Cell Adhesion and Motility

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Modelling Cell Adhesion and Motility

• An Example Invasion Wave

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Continuum Models

• Cell and chemical concentrations
• Spatial and temporal variables
• Assumed mechanisms for spatial transport
• Current models are largely phenomenological

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Spatial Transport Mechanisms

• Convection - fluid flow
• Chemotaxis - with or against chemical gradients
• Contact Inhibition - migration affected by
  cellular density
• Haptotaxis - cells move up adhesion gradients
• Diffusion - random motion down cellular gradients
• Galvanotaxis - due to electric fields

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Conservation of Massa modelling preliminary
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Flux Mass in motion

• dS is a small surface element
• N is a vector normal to dS
• J is the flux of particles. Is a vector whose
  magnitude has units of mass/area/time.

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Net Flux Through a Surface S

• To obtain the total (net) mass out of a closed
  surface S, the normal component of the flux is
  integrated over the surface
• The integral has unitsmass/time

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Concentration and Mass Flow

• If c(x,t) is a concentration at position x and
  time t then the rate of change of mass in a
  volume V bounded by S is given by the integral
  on the right.

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Thus,
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But the Divergence Theorem from calculus implies
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Integral Form of Mass Conservation

• The statement is true for any flux and any
  concentration.
• Since the volume is arbitrary the integral signs
  can be dropped

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Differential Form of Mass Conservation
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Example Diffusive Flux

• Diffusion causes particles to move in a direction
  in which the concentration is decreasing most
  rapidly
• The resulting mass conservation equation dictates
  how c(x,t) changes in time and space.

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Other Common Fluxes
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Generic Model Equations

• Most models have many fluxes of types x
• Sources and sinks F of cells/chemicals are
  included to account for processes unrelated to
  transport, such as mitosis or reaction kinetics

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ECM Mediated Tumor Invasion Waves From Physica
D 126 (1999)145-159, Perumpanani, Sherratt,
Norbury, Byrne.
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Model Assumptions

• Invading tumor cells exhibit heterotypic
  adhesion(can bind to ECM elements and other
  cells)
• Invading cells produce proteases which digest
  connective tissue
• Haptotaxis via type IV collagen - assumed to be
  spatially fixed
• Invasive cells proliferate
• Protease produced by noncontact-inhibited cells
  and are membrane bound so do not diffuse
• Protease decays linearly (and rapidly)

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Model continuum variables

• u(x,t) density of A2058 melonoma cells
• c(x,t) type IV collagen density
• p(x,t) protease concentration
• x position (single spatial variable)
• t time

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Model Equations
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Non-transport terms
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Haptotactic Flux

• Invading cell movement in direction of largest
  collagen gradient (and proportional to u)
• Unlike chemotaxis, the chemokinetic agent c does
  not diffuse.

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Numerical Simulations
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Analysis Results

• Wave speed depends on initial conditions
• For elevated collagen levels the invasion wave
  does not propagate
• Wavespeed does not depend uniquely on model
  parameters (rates etc).
• Propagation does not require random cell motility.