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9 September, 2005 A quantitative description of the invasion of bacteriophage T4 R. D. James IMA and Aerospace Engineering and Mechanics University of Minnesota Joint work with Wayne Falk Microbiology and Oral Science, University of Minnesota Thanks: John Maddocks, Rob Phillips

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9 September, 2005Warwick Bacteriophage T4: a virus that attacks bacteria Bacteriophage T-4 attacking a bacterium: phage at the right is injecting its DNA Wakefield, Julie (2000) The return of the phage. Smithsonian 31:42-6

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9 September, 2005Warwick Mechanism of infection A 100nm bioactuator This talk mainly concerns the tail sheath

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9 September, 2005Warwick Many interesting questions for quantitative analysis n How much force is generated? n Where does the energy (force x distance) come from? n How does this energy compare with similar martensitic phase transformations in materials? n What kind of elasticity theory is appropriate for a sheet of protein molecules? n …that transforms? n How is tail sheath made in Nature? n How could one make tail sheath artificially?

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9 September, 2005Warwick Elasticity theory of protein lattices Falk, James n First principles approach: protein sequence known but folded protein structure not known, interactions with solution important, time scale of transformation could be long. n Even if one could do a full, chemically accurate MD simulation for a sufficiently long time, would it answer any of the questions on the previous slide? Our approach: focus at protein level. Assume each protein molecule is characterized by a position and orientation and build a free energy.

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9 September, 2005Warwick Definitions of position and orientation Usual method (e.g., DeGennes, Chapt 2): moments of the mass distribution, e.g., 4 th moment, 6 th moment Alternative approach: Positions of atoms in the molecule Position = center of mass of the molecule Positions of corresponding atoms in the reference molecule Center of mass of the reference molecule. x y

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9 September, 2005Warwick Orientation polar decomposition Def: orientation Properties: 1) Variational characterization 2) Constrained MD simulation (linear constraint)

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9 September, 2005Warwick Protein lattice

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9 September, 2005Warwick Free energy 1) Chains: 2) Sheets: two bonding directions (i,j)-(i+1,j) and (i,j)-(i,j+1)

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9 September, 2005Warwick Frame-indifference relative orientationrelative position where “strain variables”

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9 September, 2005Warwick Compatibility Compatibility concerns the extent to which one can freely assign “strain variables”. Assume Ω is discretely simply connected. Assume that: Does there exist Does there exist such that ?

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9 September, 2005Warwick Necessary and sufficient conditions for compatibility (More bonding directions? These conditions above already determine all the positions and orientations up to overall rigid motion, which determines all other (t*,Q*) for all other bonding directions. One can write formulas for the (t*,Q*) corresponding to other bonding directions in terms of (t,Q, t, Q).)

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9 September, 2005Warwick Structure of T4 sheath 1) Approximation of molecules using electron density maps Gives orientation and position of one molecule in extended and contracted sheath one molecule of extended sheath Data from Leiman et al., 2005

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9 September, 2005Warwick Structure of T4 sheath 3) Helices II: formulas for the helices (identify (t,Q) for each) Let 2) Helices I: the 8/3 rule 3 consecutive molecules on the lowest annulus 8 consecutive molecules on the main helix For contracted sheath there is a similar 12/1 rule

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9 September, 2005Warwick Structure of T4 sheath where,, Parameters:

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9 September, 2005Warwick Values of the parameters Extended sheath: Contracted sheath: Even for homogeneous states there are lots of parameters. Is there a way to further simplify the free energy, accounting for special stiffnesses of T4 sheath?

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9 September, 2005Warwick Constrained theory I Falk, James Compute, for the main helix: The main helix : Observe: Assume above is true for all states.

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9 September, 2005Warwick Constrained theory II Now look at the rotation of molecules: Axis of rotation very near: Assume: Note: Assume: Summary: the main helix dominates

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9 September, 2005Warwick Consequence of constraint I: an unusual Poynting effect height angle

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9 September, 2005Warwick Nonuniform states transition layer by simple linear interpolation (1D ansatz) nucleation

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9 September, 2005Warwick Simplified free energy T4 sheath has three important bonding directions

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9 September, 2005Warwick Simplified free energy, continued Arisaka, Engle and Klump did calorimetry on T4 sheath, measuring the heat released on contraction. Thermodynamics + approximations:

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9 September, 2005Warwick Some linearized calculations Rigidity:

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9 September, 2005Warwick Fully relaxed states No boundary conditions, try to minimize energy by hand Try To be able to reconstruct the positions and orientations must satisfy compatibility: FRUSTRATION

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9 September, 2005Warwick Fully relaxed states Assume the minimizers of the “integrand” do satisfy Solution:Reconstructed positions and orientations: Precisely the formula for T4 tail sheath

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9 September, 2005Warwick Generic fully relaxed state Helices in every rational direction of Z 2

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9 September, 2005Warwick Force and moment at transformation.

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9 September, 2005Warwick Unrolling, interfaces One can cut tail sheath on a generator and unroll, without violating the constraints. cut unroll transform F has eigenvalues 0.567 < 2.06 “contracted” “extended”

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9 September, 2005Warwick Bio-Molecular Epitaxy (BME)

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9 September, 2005Warwick Patterning The two interfaces in the unrolled sheet: A patterning strategy:

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9 September, 2005Warwick Suggestion for producing an actuator that interacts intimately with biology, made of proteins n Biomolecular epitaxy – Lots of flexibility: molecule shape, solution chemistry n Develop patterning strategy n Test mechanical behavior n Build mathematical theory

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