Do the Ends Justify the Lengths?

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Do the Ends Justify the Lengths?

• Actin Polymer Chain Length Distribution
• Csilla Szabo
• May 22, 2006

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Overview

• What is actin?
• Actin Cycle
• Growth and decay
• Distribution of position of ends
• Length Distribution
• Joint and marginal probabilities
• Entropy

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What is Actin?

• Protein chain in cells
• Forms cytoskeleton
• Gives cells their shape
• Cell movement
• Pseudopods
• Axon growth

Leading edge of actin growth (Pollard)
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Actin Cycle
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1-D Model

• Consider a 1-dimensional model
• Let G(i, t) be the probability of finding the
  growing end at position i and time t

Time t
Position
N
N
i
i
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Growing End

• Examine chain after a given time ?t

Time t ?t
Position
N
N
i
G(i, t?t) probability of growth
probability of decay (uncap)
probability of no change
C(i, t)kuncap ?t

• G(i, t)(1 kgrow?t - kcap?t)

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Growth Continued
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Growth Continued

• Initial conditions
• Fixed initial position
• Boundary conditions
• No flux through the boundary
• Similar equations for decay and capping
• Solve system of PDEs numerically using Explicit
  Eulers Method

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The Question

• Given the distribution of the position of growing
  and decaying ends of the chain can we find the
  length distribution of the chains?

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Joint Probabilities
N
i
j

• Pij probability of having the growing end at
  position j and the decaying end at position i

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Joint Probabilities
Position of decaying end
Position of growing end
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Marginal Probabilities
gj P1j P2j P3j Pj-1j
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Marginal Probabilities
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5 Node Case
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Equations for 5 Node Case
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Entropy

• Measure of randomness or uncertainty in the
  system

• Maximize entropy

• Set 0

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Rewrite System

• Goal Find all probabilities in terms of known
  quantities
• Use the new equation for Pij?P1jPiN/P1N
• Algebra magic
• Yuck!! More work to be done! This is a simple 5
  node case, most simulations have 100 nodes

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Test Case

• 10 chains one of each type
• Pij 1/10
• Equations hold for this case!

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Questions?