Model results (Actin Dynamics)
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This document summarizes results from models of actively contracting bundles of polar filaments and bipolar filaments. It finds that: 1) Instability in bundles of polar filaments is due to interactions between parallel filaments, and the critical value (αc) above which the system is unstable increases with diffusion coefficient (D) and decreases with system size (L) and treadmilling rate. 2) For bundles containing both polar and bipolar filaments, αc decreases with the splitting rate (wd) of bipolar filaments and increases with the diffusion coefficient (D). The relationship between αc and other parameters like system size (L) and bundling rate (β) is more complex.
![Results and Observations
As we have seen earlier that for bundles of small size have bigger αc i.e. shortening of bundles lead to
inhomogeneous stable state(generally happens beacause of unstable homogeneous state).As we
can see tfrom previous results that these unstabilities occur due to more filament interactions
which leads to shortening.However for shortening of these bundles,someone has to perform
mechanical work which is coming from these interactions in terms of stress produced by forces
generated by motors and hydrodynamic forces of these fluids.If we ignore interaction b/w different
filaments due to friction force,considering filament to be rigid-rod like filamentmoving in a viscous
environment with velocity v and mobility μ ,if there is a force on filament at point smot ,then we can
write explicitly force balance neglecting mass momentum of rod,
∂sσ =fmot + v/μ
After using this knowledge ,we can write it's tension profile which piecewise linear.
Image Source: K. Kruse
Stress is positive in the rear end as it is being pulled and negative as it is being stretched.To get total stress
profile along the bundle, we need to sum all stresses in all filaments.Using above stress profile,we calculate
an average stress profile along a filament a x by summing over all contributions due to interactions.We then
can calculate stress in bundle at position y,by summing over average stress of filaments with their '+'ve end
within the certain interval (for '+' ve actin filament [y-Ɩ ,y],for '-'ve actin filament [y,y+ Ɩ] & for bipolar
ones [y- Ɩ,y+ Ɩ]).](https://image.slidesharecdn.com/modelresults-151007115614-lva1-app6892/85/Model-results-Actin-Dynamics-6-320.jpg)
Model results (Actin Dynamics)
- 1. Results and Observations Model considering actin polar filaments This model has already been discussed in “Actively Contracting Bundles of Polar Filaments”,PRL (Vol. 85 No. 8).However the model that we have used to describe dynamics includes treadmmilling (current: ∂x(vtrc+- ) ).So In this model as it's straightforward to see from the graph that αc >0 for all parameters (when vtr=0), which means unstability is due to interaction between parallel filaments with α≥ αc Also it can be easily seen that αc increases with increasing D, diffusion coeff.(As (magnitude only) in graph (is actually β*Ɩ/D where Ɩ is length of filament (both filaments were assumed of same length)) is decreasing , αc increases).As soon as we decrease size of system L, αc increases(shortening of bundles gives stability).Also αc decreases with increasing c22 w.r.t. i.e. increasing filaments of same orientation.(graph 1)Also When treadmilling is accounted,αc decreases which means treadmilling is also responsible for destabilization of system.Now it might be said for that region when αc decreases with increase in D is due to dominance of treadmilling in those region.(see graph 2). Graph 1 (Here coordinates on X-axis is and coordinates on Y-axis is .Curves with green color indicate more fraction of c22 compared to c11 and and red color indicate lesser fraction c22 compared to c11.)
- 2. Results and Observations Graph 2(Here in this case c11=.3 and c22=.7,Here coordinates on X-axis is and coordinates on Y-axis is ) Model considering actin polar filaments and Bipolar filaments This model is an extension of previous model with consideration of Bipolar filaments which has been mentioned in "Self-organization and mechanical properties of active filament bundles", Physical Review E 67 051913. However the model that we have used to describe dynamics includes treadmmilling (current: ∂x(vtrc+- ) ). Also it can be easily seen that αc decreases with increasing wd ,rate of splitting of bipolar filaments to result in polar ones.(As (magnitude only) in graph (is actually β/wd*Ɩ where Ɩ is length of filament (both filaments were assumed of same length)) is decreasing , αc decreases).As soon as we decrease size of system L, αc first decreases then increases(shortening of bundles for stabiltiy now has barrier ).Also αc may increase with increase in β(not always! In green curve,for β (negatvie part), αc has very random patterns) and sometimes function is multivalued, So no certainity for all α<αc is stable region , while in previous model (only polar filaments) , αc decreases with increase in β and with certainity. So In this model, more anti parallel filament interactions causes stability to system(as β increases,more stabiltiy to system).Also αc decreases with increasing c22 w.r.t. i.e. increasing filaments of same orientation.It stabilizes(αc increases) with increase in D,diffusion coefficient.
- 4. Results and Observations Constant Prameters: (C0 + =.3 , C0 - =.7, D=2, β=.5 ,wd=.7 ,wc=.3 ,Ɩ=1) Also αc decreases with increase in wc ,So combination of polar filaments(antiparallel) resulting in bipolar filament causes unstability to system( wc is rate of combination of polar filaments).If this result is in consistence with other observations, then it can be understood physically that "As antiparallel filaments may cause stability to system,so making of bipolar filaments by combination will bring unstability to system."(however this argument seems to be mathematically wrong when wc<wd ,as for these values of wc ,we get polar filaments efectively as wc>wd which is not happening as you can see from graph curve with lesser β has lesser stability for all times.By looking at graph ,one thing that is needed to be observed,for system to have stability with lesser L (contraction) is not so straightforward.as it can be seen that for wc lesser than certain value,system has more stability than system of lesser size(really ???).
- 6. Results and Observations As we have seen earlier that for bundles of small size have bigger αc i.e. shortening of bundles lead to inhomogeneous stable state(generally happens beacause of unstable homogeneous state).As we can see tfrom previous results that these unstabilities occur due to more filament interactions which leads to shortening.However for shortening of these bundles,someone has to perform mechanical work which is coming from these interactions in terms of stress produced by forces generated by motors and hydrodynamic forces of these fluids.If we ignore interaction b/w different filaments due to friction force,considering filament to be rigid-rod like filamentmoving in a viscous environment with velocity v and mobility μ ,if there is a force on filament at point smot ,then we can write explicitly force balance neglecting mass momentum of rod, ∂sσ =fmot + v/μ After using this knowledge ,we can write it's tension profile which piecewise linear. Image Source: K. Kruse Stress is positive in the rear end as it is being pulled and negative as it is being stretched.To get total stress profile along the bundle, we need to sum all stresses in all filaments.Using above stress profile,we calculate an average stress profile along a filament a x by summing over all contributions due to interactions.We then can calculate stress in bundle at position y,by summing over average stress of filaments with their '+'ve end within the certain interval (for '+' ve actin filament [y-Ɩ ,y],for '-'ve actin filament [y,y+ Ɩ] & for bipolar ones [y- Ɩ,y+ Ɩ]).