Re-entrant transition in a one-dimensional growth model

Transcription

1 Re-entrant transition in a one-dimensional growth model Ostap Hryniv Dept. of Mathematical Sciences Durham University Ostap.Hryniv@durham.ac.uk X International Conference Stochastic and Analytic Methods in Mathematical Physics Yerevan, Armenia, September 4 11, 2016

2 Outline Model Motivation Heuristics Results and Ideas Conclusions (partially based on a joint work with A. Letcher and D. Sheard)

3 The model State space long polymer chains made of and monomers Dynamics: Growth: attachment of a monomer:... =... rate λ +... =... rate λ Conversion: independent transformation : = rate 1 Shrinking: detachment of the extreme monomer:... =... rate µ Want: long term behaviour of the process.

4 Motivation relatively simple growth model, only two types of particle, three parameters a version of a model suggested in [Antal et al PRE 2007] to describe microtubule dynamics wild fluctuation under steady state; rich dynamics within a simple soluble model; various approximations via the Master equation a non-reversible model exhibiting metastable behaviour; interesting phase transition

5 What are microtubules? Microtubules are one of the components of the cytoskeleton. They have a diameter of 25 nm and length varying from 200 nm to 25 µm. [ [Howard & Hyman, Nature 2003]

6 Microtubule structure and dynamics Microtubules are polymers made of protofilaments organised into an imperfect helix. Protofilaments, in their turn, consist of α- and β-tubulin dimers. [Howard & Hyman, Nature 2003]

7 Microtubule dynamics During polymerization, both the α- and β-subunits of the tubulin dimer are bound to a molecule of GTP. While the GTP bound to α-tubulin is stable, the GTP bound to β-tubulin may hydrolyze to GDP after assembly. The GDP-tubulin leads to depolymerization of the tubulin dimers at the microtubule tip: [Howard & Hyman, Nature 2003]

8 Single microtubule dynamics Growth: µm/min (ie., dimers/sec); shrinking about 10 times faster: The model mimics the experimental data [Fygenson et al, PRE 1994] [avi]

9 Heuristics Microtubles long chains made of and monomers (guanosin triphosphate (GTP + ) and guanosin diphosphate (GDP ) tubulin complexes) Microtubules (position, head), ie., a Markov process y t (x t, w t ), t 0 ; The head process (w t ) t 0, is a non-reversible Markov chain in W = { } { w = w, where w is a finite word of and } State is recurrent for w t ; leads to a useful renewal structure. [alternating -cycles and -cycles]

10 Empty-head process Let 0 = τ 0 < τ 1 <... be times of consecutive returns of w t to. def Put ỹ l y τl = (x τl, ) with x l x τl, a CTRW in Z. Key facts: [HM 10, H 12] The space-time increments of x l have exponential moments in a neighbourhood of the origin; [cluster expansion lemma!] Many classical results (SLLN, CLT, LDP,... ) hold; The velocity v of x t is given by x t v lim t t = E x 1/E τ 1 (a.s.) ; it is analytic in λ +, λ and µ. Phase boundary {v = 0} separates the compact phase (v < 0) from the region of indefinite growth (v > 0).

11 Explicit expression for v? w t irreducible, positive recurrent MC in W with stat. dist. π; put π + = π ( {w W : w 0 = } ), π = 1 π +. Write v + λ + > 0 and v λ µ. By the Ergodic theorem, x t v = lim t t = π + v + + π v (a.s.). If λ > µ (ie., v > 0), then v > 0 and x t +. However, no good control of π + (π ), but finite-window approximations for π give v > 0 for µ λ (1 + λ + ). [HM 10] asymptotically correct for small λ ± [Antal et al PRE 2007]

12 Lifetimes (region v < 0) Eventually, every monomer detaches from the microtubule; let T min { t > 0 : y t = ( 1, ) y 0 = (0, ) }, T min { t > 0 : y t = ( 1, ) y 0 = (0, ) }. Clearly, T stochastically dominates T. Can show 1 + µ ET = λ ET ; so that ET < iff ET < ; moreover, for small s > 0, v < 0 E T < E e st <. Actually, in this region, one has v = 1/ET < 0. Can also relate ϕ (s) def = Ee st and ϕ (s) def = Ee st. [HM 10]

13 Further results Regular phase Theorem [H 12] Let λ λ +. Then v is a strictly monotone function of the rates λ +, λ, and µ, with partial derivatives v λ + > 0, v λ > 0, v µ < 0. [natural monotonicity properties] Non-strict monotonicity: via coupling. Strict monotonicity: via large deviations and the Ergodic theorem.

14 Numerics Re-entrant phase transition Microtubule velocity; Lam+ = 10 Lam- = 0.01 Lam- = 0.03 Lam- = 0.05 Lam- = Velocity Mu

15 Microtubule velocity; Lam+ = 10 Lam- = 0.01 Lam- = 0.03 Lam- = 0.05 Lam- = Velocity Mu

16 Re-entrant phase transition Theorem: Fix λ + large and λ small. Then v = v(µ) changes sign three times. For µ small : if µ < 1, the -cycle often stops once the extreme monomer hydrolyzes, ; there is exactly one intersection point (v = 0) with µ = O(λ + ).

17 Re-entrant phase transition Large µ heuristics: if µ = +, x t is an additive functional of the classical birth-and-death process; gives complete control of the asymptotic behaviour; if µ < but still large, use cluster expansions around the µ = case (λ contribution is still small); the argument relies upon the statistical homogeneity of the trajectories of y t at the microscopic level and sharp control of the metastable behaviour of the related birh and death process.

18 Conclusions The model behaves naturally for λ λ + ( regular phase), while for some pairs (λ +, λ ) with λ < λ + a re-entrant transition is observed (faster shrinking rate can increase the velocity); Description of the whole region (λ, λ + ) exhibiting the re-entrant transition still lacking; More advanced models in the spirit of [Piette et al 2009] are of interest (eg., allow the extreme to detach; some results are available);...

19 References T.Antal, P.L.Krapivsky, S.Redner, M.Mailman, and B.Chakraborty. Dynamics of an Idealized Model of Microtubule Growth and Catastrophe. Phys. Rev. E76, (2007). B.M.A.G.Piette, J.Liu, K.Peeters, A.Smertenko, T.Hawkins, M.Deeks, R.Quinlan, W.J.Zakrzewski, and P.J.Hussey. A Thermodynamic Model of Microtubule Assembly and Disassembly. PLoS ONE 4(8): e6378 (2009). OH, M.Menshikov. Long-time behaviour in a model of microtubule growth. Adv. Appl. Prob. 42, (2010). OH. Regular phase in a model of microtubule growth. Markov Processes and Related Fields 18(2): (2012). OH, A. Letcher and D. Sheard. Re-entrant phase transition in a model of microtubule growth. [in preparation]