Lecture III Dynamic Instability and the uses of energy

The boundary conditions for microtubule growth: the risk of

rogue microtubules

In animal cells microtubules are nucleated and grow randomlyfrom the centrosome

Fragments of microtubules could lead to much random and non-productive polymerizaton

J

ce

J

cc

cc

In nucleated microtubules only the plus or end is active. The overall growth rate of the microtubule is given as J. For an equilibrium polymer, J is always positive above ce, and always negative below cc. For dynamic instability between cc and cc

free microtubules shrink but nucleate microtubules with free ends continue to grow. The steady state concentration in a cell where there are nucleated microtubules is cc

cc

J

J

Bounded and Unbounded GrowthIn an equilibrium polymer unbounded growth occurs above the critical concentration and no growth occurs below the critical concentration. However in dynamic instability a nucleation site will contain polymer below the critical concentration and the amount of polymer will reach a steady state value.We can calculate the average length as a stochastic process from just four constant parameters: vg (the continuous rate of growth), vd (the continuous rate of shrinkage), fcat (the frequency of going from growth to shrinkage also known as catastrophy), and fres the frequency of going from shrinkage to growth, also known as rescue. The probability that a microtubule is of length L at time t and is in the growing state is pg (L,t) and also for shrinkage, ps (L,t)

L-1 L L L+1

g

g ss

vg/lvs/l

fcat

fres

The major proposal for the purpose of dynamic instability was that it enabled the placement of filaments in the cell by a process of variation and selection. Holy and Liebler (PNAS, 1994) asked whether the selection part of variation and selection was optimized in the case of dynamic instability. Was this as they asked an effective process for morphogenesis. This is what may be one element of perhaps Leibler’s major contribution to systems biology, the question of optimization of a process, and the further question of what it was optimized for: robustness, speed, efficient use of energy, versatility, etc. In this case the test was for speed of morphogenesis.

versus

The behavior of an equilibrium polymer depends on the on-rate, ron and the off rate, roff. For a dynamic instability it depends, as we saw on four parameters: vg, vs, fcat, and fres. Changes in an equilibrium polymer are due to fluctuations on the ends and diffusion of the ends or the target to each other. For DI the ends fluctuate due to rapid polymerization depolymerization. Which is more efficient.

Consider a radial pattern of microtubules which must interact with chromosomes at a distance d from the point of nucleation.

A chromosome diffuses very slowly taking about 1000 minutes to move between one microtubule and another. An equilibrium polymer would be stable in that region for about that time but for a microtubule showing dynamic instability the time would be about minute. Hence diffusion would play a small or negligible role. See Holy and Leibler PNAS (1994)

In the bounded state the length distrubtion is exponential and at steady state it is given by a simple expression:

If one calculates search times they are optimum when d = <L> and fres=0

What really happens is that instead of small fluctuations the growing and shrinking polymers explore more space.

The amount is simply given by the relative growth in a given time. This scales for fluctuations at the ends of an equilibrium polymer to the ratio of the monomer size and the length of growing microtubules in the case of dynamic instability.

According to Holy and Leibler the parameters of microtubule are optimized for this search process. For a number of cells the fres= 0, which means the length of the growing microtubule is about d. The time of capture is about that predicted by theory.

This can be simulated to show dynamic instability and bounded growth: http://www.embl-heidelberg.de/~nedelec/other/dynamic_aster/index.html

In addition to dynamic instability motor proteins play a big role in morphogenesis. Reconstructions in vitro show their amazing versatility

The myosin family(actin filaments)

The kinesin family (microtubules)

From Surrey et al. Science (2001) 292 1167

Effect of crosslinked kinesin in vitro

Both actual experiments and simulations agree on general morphogenetic properties of oligomeric motor proteins, In experiments by Nedelec’s group this can be seen graphically. http://www.cytosim.org/reprints/self2/index.html

Theoretical effects of motor density

Can simple rules for behavior generate patterns in much more

complex systems?

Stigmergy and self-organization

Principles by which simple rules interact with the environment, change the environment, and lead to higher level organization.

Worker ants in a few hours take corpses and arrange them into regularly placed piles

The rules for this have been worked out by Theraulz et al., PNAS 2002. The custers are dynamic and grow and shrink . There are several rules that govern this:

Rules for cemetary construction

• The probablility of picking up a corpse decreases with cluster size

• The probability of dropping a corpse increases with cluster size

• The depletion of corpses reduces the pick up rate, a form of negative feedback

These behaviors can be described by two differential equationsHere c(x,t) is the density of corpses and a(x,t) is the density of corpse-carrying ants, v is the velocity of the ants, is the density of non-laden ants, D is a diffusion term of the ants in two dimensions, and is a term for short-term interactions within the space perception of the ant. The dropping rate increases with but the picking up rate decreases with :

Key features of the behavioral model and its relation to

biochemical models• Structure emerges in a homogeneous condition

due to positive and negative feedback• Multistability arises through non-linearity of

equations and stochastic behavior• Bifurcation spatially is very density dependent.

This requirement for a certain density of individuals is seen in many examples of ant and termite behavior

Simulations demonstrate that these rules generate multiple

clumps from random distributions:

http://www.jweimar.de/jcasim/acri_ants.html

Take home lessons

• Simple rules (microtubules; ants) use simple properties to interact with the environment to give patterns

• In ways not descriped the patterning mechanism both responds to and alters the environment

• The patterns are robust to changes in the environment and adaptive to the environment

References

• Kirschner MW Implications of treadmilling for the stability and polarity of actin and tubulin polymers in vivo.J Cell Biol. 1980 Jul;86(1):330-4.

• Verde F, Dogterom M, Stelzer E, Karsenti E, Leibler S.Related Articles, Links Control of microtubule dynamics and length by cyclin A- and cyclin B-dependent kinases in Xenopus egg extracts.J Cell Biol. 1992 Sep;118(5):1097-108.

• Holy TE, Leibler S. Dynamic instability of microtubules as an efficient way to search in space.Proc Natl Acad Sci U S A. 1994 Jun 7;91(12):5682-5.

• : Surrey T, Nedelec F, Leibler S, Karsenti E. Physical properties determining self-organization of motors and microtubules.Science. 2001 May 11;292(5519):1167-71.

• : Franks NR, Pratt SC, Mallon EB, Britton NF, Sumpter DJ. Information flow, opinion polling and collective intelligence in house-hunting social insects.Philos Trans R Soc Lond B Biol Sci. 2002 Nov 29;357(1427):1567-83. Review.

• Theraulaz G, Bonabeau E, Nicolis SC, Sole RV, Fourcassie V, Blanco S, Fournier R, Joly JL, Fernandez P, Grimal A, Dalle P, Deneubourg JL. Spatial patterns in ant colonies.Proc Natl Acad Sci U S A. 2002 Jul 23;99(15):9645-9. Epub 2002 Jul 11.

Force and polymerization: breaking symmetry

We will start with demonstrating a relationship and then demonstrating

significant biological importance

Effect of actin polymerization on lipid vesicles coated with Act A

1: Proc Natl Acad Sci U S A. 2003 May 27;100(11):6493-8. Epub 2003 May 8. Related Articles, Links

controlActA

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Sickle cell Anemia

Even without nucleotide hydrolysis polymers like actin and microtubules can exert force and do work

We consider the situation where the thermodynamic work that is done is pushing with a force, F, over a distance of one subunit that is inserted (l0 = L/N). For a microtubule which is 13 strands l0 is 8nm/ 13 = 0.615nm

We assume that the force obeys Hooke’s law whereF = a(l - l0)l = F/a +l0

F

The chemical potential is given by dldF (at constant temperature) dF/a + l0)dFOn integrating:l0F + (F2/2a)for small Fl0F

Extension F>0

Now considering the free monomers in solution. At equilibrium their chemical potential is given by:

s0 + kT ln ce

0

l0F s0 + kT ln ce

Combining these we get that: lnce = lnce

0 - (l0F)

(solution)

+ end

- end

J

ce

0

Jc -

Jc -

F

ce

’

The reciprocal effect of force on the rate constants

+ end

F

’

Capped ’=0

Consider the situation at equilibrium.ce

0 = ’/

ce = ’/

kTln(/’s

0 -

kTln(/’s0 -l0F

There is a relationship between the rate constants’/ = /

’ elF/kT

The difference in free energy is shared between the two rates

eflF/kT

’ef-lF/kT’We might expect that f is itself a function of F. In any case compressive force should make it difficult to add subunits; the higher the force the more difficult. Experimental studies show that compressive force only affects on-rate.

Does polymerization and depolymerization do important work?

The first case was the demonstration that the depolymerization of microtubules can pull chromosomes.

This led to statistical mechanical models to ask whether the thermodynamic feasibility could be achieved. Since the process depends on the diffusion of the filament away from the wall, the process could either be diffusion limited or reaction limited. By diffusion limited one assumes that there are a high concentration of subunits ready to jump into the gap but they wait for a rare movement. By reaction limited, the fluctuations are high but the not every fluctuation inserts a subunit. This process is called a Brownian Ratchet after the machine described by Richard Feynman

Pulling the microtubules by microtubule depolymerization: The Dam 1 Ring complex.Formation of a dynamic kinetochore- microtubule interface through assembly of the Dam1 ring complex.Westermann S, Avila-Sakar A, Wang HW, Niederstrasser H, Wong J, Drubin DG, Nogales E, Barnes G. Mol Cell. 2005 Jan 21;17(2):277-90

Following the GTP subunits

Force-dependent growth and Symmetry Breaking

QuickTime™ and aTIFF (Uncompressed) decompressor

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The listeria assay for polymerization driven motility.

The active protein in listeria, ActA, can be adsorbed to beads and actin will assembly.

Initially the there is a cloud of actin around the bead which establishes no strong polarity. Suddenly wihtin a few seconds it becomes asymmetrical and becomes motile.

QuickTime™ and aTIFF (LZW) decompressor

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Movie of Symmetry Breakinghttp://web.mit.edu/biophysics/movies/bead4.mov

The utility of rigid mechanical structures in symmetry breaking.Symmetry breaking or large distances (eg. in eggs and embryos is difficult. It is hard to repress local autonomy. In the case of a bead of Act A, how is one side to be repressed and the other activated?

QuickTime™ and aTIFF (LZW) decompressor

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Polymerization is nucleated by Act A on the bead so if the filament too far from the bead polymerization is low (A) but if it is too close to the bead the force on the actin polymerization slows the polymerization (B).

AB

The effect of the bead is to correleate the forces. For example, two filaments next to each other will stimulate the polymerization, since the force acting on one will decrease the force acting on the other. For filaments on opposite sides of the bead the actions will be anticorrelated

QuickTime™ and aTIFF (LZW) decompressor

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As we saw earlier the on-rate is diminished by :

’exp(-f kT)

The model also explains the large lag (over one hour) before symmetry is broken.

1. In the first stage actin filaments assemble there is little force exerted.

2. In the next stage force builds up symmetrically on the bead. This increases the off- rate. At a critical off rate, the sotochastic nature of the asembly is magniifed

3. Correlated unidirectional growth follows.

Give story of Xenopus polarity and selforg

Perhaps with some other stories of how eggsDo it Nematode, Drosophila Daniel stjohnston