Ring closure in actin polymers

0.14 MB·

by  Supurna Sinha

Ring closure in actin polymers
Supurna Sinha 1 and Sebanti Chattopadhyay 2
1Raman Research Institute, Bangalore 560080, India.
2Doon University, Dehradun 248001, India.
(Dated: January 16, 2017)
We present an analysis for the ring closure probability of semiflexible polymers within the pure
bendWormLikeChain(WLC)model. Theringclosureprobabilitypredictedfromouranalysiscan
betested against fluorescent actin cyclization experiments.Wealso discuss theeffect ofringclosure
on bend angle fluctuationsin actin polymers.
7
1
PACSnumbers: 87.15.Cc,62.20.de,82.35.-x,87.10.-e
0
2

n I. INTRODUCTION method for solving the wormlike chain model for semi-
a flexible polymers to any desired accuracy over the entire
J In the past two decades, there has been much interest range of polymer lengths to determine Q˜(~r). The plots
1 in the theoretical study of semiflexible polymer elastic- for Q˜(~r) for various β = L , the ratio of the contour
1 LP
ity. These studies are motivated by micromanipulation length L to the persistence length LP, reveal the depen-
] experiments[1–3]onbiopolymers. Inparticular,inrecent dence ofthe endto enddistance vectoronthe rigidityof
t
f years there have been experiments involving stretching the polymer (See Fig. [4] in [7]).
o DNA molecules[1] which give us information about the We outline the theoretical calculation of Q˜(~r) below.
s
. bend elastic properties of DNA. There have also been (For a detailed exposition please see Appendix A). Con-
t
a experiments on fluorescently tagged actin filaments[4] sider a situation where the initial and final tangent vec-
m wheretheymeasurethebendpersistencelengthofactin. tors(tˆA = dd~xs|s=0 andtˆB = dd~xs|s=L)areheldfixed. Then
- More recently, there have been fluorescence experiments Q˜(~r) has the following path integral representation:
d
on cyclization of actin filaments[5]. In these papers they
on analyze the formation of rings in actin polymers and Q˜(~r) = N D[tˆ(s)]exp{− 1 A L(dtˆ)2ds }
c study the effect of ring closure on bend angle fluctua- Z kBT 2 Z0 ds
[ tions in these polymeric rings. Our interest here is lim- L (cid:2) (cid:3)
1 ited to the process of cyclization itself and therefore in ×δ3(~r− tˆds) (2)
v our analysis we restrict to polymers with only bend de- Z0
0 grees of freedom and no twist degree of freedom. Actin HereN isthenormalizationconstantandkBT,thether-
2 cyclization is of interest to biologists [6] who do visual- mal energy at temperature T. As mentioned in [7], we
5 ization studies of actin ring formation in the context of solve for Q˜(~r) by first considering a related end to end
3
0 cell division. distance measure:
.
1 P(z)= d~rQ˜(~r)δ(r −z),
0 3
II. RING CLOSURE PROBABILITY
7 Z
DISTRIBUTION
1 whichisQ˜(~r)integratedoveraplaneofconstantz. This
v: inturnis relatedtoP˜(f), the LaplacetransformofP(z)
Our starting point is the pure bend Worm Like Chain
i given by:
X (WLC) model[7]. In this model, the polymer configura-
r tion is viewed as a space curve ~x(s). There is a tangent L
a vectorassociatedwith eachpointonthe polymerofcon- P˜(f)= P(z)eLfPz dz (3)
tour length L and the energy of configuration is given Z−L
by: f, the variable conjugate to z has the interpretation of
A L a stretching force and thus P˜(f), can be written as the
E[C]= dsκ2 (1) ratioZ(f)/Z(0)ofthepartitionfunctionsinthepresence
2
Z0 and absence of an external stretching force f. We do an
where C stands for the polymer configuration. A is the eigenspectrumanalysisofP˜(f)anddetermineQ˜(~r)using
bending elastic constant and the curvature κ=|dtˆ|. tomographic transformations outlined in [7].
ds
One of the key quantities characterizing the elasticity Hereweaddressaquestionwhichisofcurrentinterest
of a biopolymer is Q˜(~r), the probability distribution for to application of polymer physics to biology: cyclization
the end to end distance vector ~r between the two ends of actin filaments[5]. Within the pure bend Worm Like
of the polymer as it gets jiggled around by thermal fluc- Chain (WLC) Model we compute the ring closure prob-
tuations in a cellular environment [7]. In [7] we use a ability (RCP) by considering Q˜(~r =~0).
2
III. METHOD
In Fig. 4 of [7] we display a family of curves of Q(ρ)
versus ρ, with ρ = |~r| for various values of β. Q(ρ) is
β
a theoretically convenientquantity expressedin terms of
scaledunits(ρ~= ~r). Inordertocomputetheringclosure
β
probabilitydensityQ˜(~r =~0)weneedtochangevariables
from ρ= |~r| to |~r|=r. Setting Q˜(~r)=Q , we get:
β ~r
Q(ρ~)dρ~ = Q d~r (4)
~r
Z Z
FIG. 1: A plot of the ring closure probability density Q(ρ~=
or
~0)= Q(0) versus β, setting LP =1. It has a small value for
shortpolymerswhich arehard tobendandform ringsandit
has a large value for long polymers which are easy to bend
Q(ρ~)
d~r = Q d~r (5) and thustheprobability density of ring formation is high.
β3 ~r
Z Z
which in turn implies
Q(0)
= Q (6)
β3 0
We compute Q(0) for a range of values of β using Math-
ematica. As we can see from the plot of the ring closure
probabilitydensityQ(0)versusβ (Fig. 1),thatQ(0)has
a small value for short polymers which are hard to bend
and form rings and it has a large value for long poly- FIG.2: Aplotoftheringclosureprobabilitydensityinphys-
mers which are easy to bend and thus the probability ical space Q0 = Q(0)/β3, versus β setting LP = 1. Notice
density of ring formation is high. We then compute and thatthisfunctionissmallforverysmallandlargeβandpeaks
around an intermediate valueβ ≈3.
plottheringclosureprobabilitydensityinphysicalspace,
Q = Q(0) as a function of β (Fig. 2). The qualitative
0 β3
features of the plot shown in Fig. 2 are in agreement modes which do not contribute, we find that the contri-
with our intuition. The ring closure probability density bution from the x−y plane is given by
Q in physical space, which is an experimentally mea-
0 1 6 L
surable quantity is small for very short and long strands <φ2 >xy = (1− ) (7)
ring 12 π2 L
of the polymer and peaks around intermediate contour P
lengths of L≈3LP (See Fig. 7-41 on page 438 of [8]). We need to add this contribution to the contribution
coming from the z direction where the ring closure con-
dition is of the form
IV. MEAN SQUARED TANGENT ANGLE L
FLUCTUATION φ (s)ds=0.
z
Z0
One of the experimentally relevant quantities of inter- In this case the Fourier expansion for φz(s) can be
est is the mean squared tangent angle fluctuation[5]. In expressed as φz(s) = ∞n=2φne2πLins which finally gives
Ref. [5] the mean squared tangent angle fluctuation has us
P
been calculated for a ring and a linear filament in a two 1 6 L
dimensional setup. They find good agreement with ex- <φ2 >zring= 12(1− π2)L (8)
P
perimental measurements.
Thus combining Eqs. 7 and 8, the net mean squared
Herewepresentasimilarcalculationinathreedimen-
tangent angle fluctuation for a three dimensional ring is
sional geometry. Consider a polymer configuration in a
given by
closed circular ring lying in the x−y plane. Expanding
the bend angle fluctuation φ(s) in a Fourier series and 1 6 L
<φ2 >3d = (1− ) (9)
imposing the ring closure constraint and removing zero ring 6 π2 L
P
3
the mean squared tangent angle fluctuation to be tested
against future experiments on fluorescently tagged ring
like and linear actin filaments in a three dimensional ge-
ometry.
ACKNOWLEDGEMENT
One of us (SC) would like to thank RRI for providing
hospitalityduringstayinRamanResearchInstitute asa
Visiting Student.
FIG. 3: Plots of mean squared tangent angle fluctuation in APPENDIX A: COMPUTATION OF Q0
threedimensionsforaring(dashedline)andalinearfilament
(solid line). We have set LP = 1. Notice the suppression of
OurgoalistocalculateQ ,theringclosureprobability.
fluctuation for a ring filament compared to a linear one. 0
Q is Q˜(~r) for ~r =~0. Q˜(~r), the probability distribution
0
fortheendtoendvector~r forasemiflexiblepolymerhas
Asimilarcalculationforalinearfilamentinthreedimen- the following path integral representation:
sions gives us
1 A L dtˆ
Q˜(~r) = N D[tˆ(s)]exp{− ( )2ds }
<φ2 >3lidn= 31LLP (10) Z L kBT(cid:2)2 Z0 ds (cid:3)
×δ3(~r− tˆds) (11)
Wehaveplotted(9)and(10)inFig. 3. Thesepredictions
Z0
canbetestedagainstfutureexperimentsonfluorescently
HereN isthenormalizationconstantandk T,thether-
taggedactinfilaments. Noticethat,asinthetwodimen- B
mal energy at temperature T.
sional case[5], we find that < φ2 > is suppressed for a
InsteadofQ˜(~r)itturnsouttobeeasiertofirstconsider
ringlike structure compared to a linear filament. This
P(z)
indicates that cyclization is entropically costly. Also, as
expected, the fluctuations are smaller in the two dimen-
P(z)= d~rQ˜(~r)δ(r −z),
sional geometry compared to the three dimensional ge- 3
ometry. Z
which is Q˜(~r) integrated over a plane of constant z.
P(z)in turn is relatedto P˜(f), the Laplacetransform
V. CONCLUSION of P(z) given by:
L
Our treatment is an analysis based on the pure bend P˜(f)= P(z)eLfPz dz (12)
Worm Like Chain Model. The absence of twist degree Z−L
of freedom enables our predictions to be directly tested
f, the variable conjugate to z has the interpretation of
againstactincyclizationexperimentswherethetwoends
a stretching force and thus P˜(f), can be written as the
of the polymer come together without any relative twist
ratioZ(f)/Z(0)ofthepartitionfunctionsinthepresence
between the two ends. This is to be contrasted with
and absence of an external stretching force f.
analysisof J factor of DNA with twist degreeof freedom
Z(f) is given by
where the additional twist degree of freedom makes the
analysis considerably more cumbersome[9–11]. In [11]
aninterpolationformulaispresentedinthe intermediate L L dtˆ 2
rigidity regime. They [11] also presented analytical ex- Z(f)=N D tˆ(s) exp − P ds
2 ds
pressions for the ring closure probability density Q in Z ( "Zo (cid:18) (cid:19) #)
0 (cid:2) (cid:3)
the limit of β <<1 and β >>1. However, they did not f L
×exp tˆds
have an exact expression for the entire range of polymer L 2
" P Z0 #
lengths. In contrast, here we present a semianalytical
study which gives an essentially exact prediction for Q which in turn can be expressed as
0
fortheentirerangeofrigidity(See Fig. 1andFig. 2). It
would be interesting to see how our predictions quanti-
β 1 dtˆ 2
tativelycomparewithfuturecyclizationprobabilitydata Z(f)=N D tˆ(τ) exp − dτ −ftˆ
z
for actin filaments. We also expect our predictions for Z ( Zo "2(cid:18)dτ(cid:19) #)
(cid:2) (cid:3)
4
which has the interpretation of the kernel of a quantum Z=M[[1,1]];
particle on the surface of a sphere at an inverse tem- L=Append[L,Z]]
perature β. We now exploit the analogy between time L=Re[L];
imaginary quantum mechanics and classical statistical Pz={};
mechanics to re-express Z(f) as follows: P1z={}
For[l=-2,l¡1200,l++,
xi=.001*l;
Z(f)= e−[βEn]ψn tˆA ψn tˆβ P=(h*beta/Pi)*Sum[L[[n]]*Cos[(n-
Xn (cid:0) (cid:1) (cid:0) (cid:1) 1)*h*xi*beta],n,1,Nmax];
where{ψ tˆ }isacompletesetofnormalizedeigen- Pz=Append[Pz,xi,P];
n A
P1z=Append[P1z,P]];
statesoftheHamiltonianH =−∇2−fcosθandE are
(cid:0) (cid:1) f 2 n V=P1z;
the corresponding eigenstates. For free boundary condi-
QR1=;
tions for the end tangent vectors we can express Z(f)
L1=Drop[V,2]
as
L2=Drop[V,-2]
LPR=(L1-L2)/(.001*2);
Z(f)=ho|exp−βH |oi
f
LPR=Drop[LPR,1];
TheHamiltonianH =−∇2−fcosθ istheHamiltonian (*Computation of S(r)*)
f 2 QR = Table[LPR[[i]] ∗ 1/((i − 1) ∗ .001) ∗ [−1/(2 ∗
of a rigid rotor in a potential and |0 > is the ground
stateofthefreeHamiltonianH =−∇2. Wenumerically Pi)],i,2,1199];
0 2 (*Computation of Q(r)*)
evaluate Z(f) by choosing a suitable basis in which H
f QR1=Table[(i−1)∗.001,LPR[[i]]∗(1/(i−1))∗
has a tridiagonal symmetric matrix structure with
1/((i−1)∗.001)∗[−1/(2∗Pi)],i,2,2];
ListPlot[QR1]
l(l+1)
H =
ll
2
Program for computing Q0
H =f(l+1) 1/[(2l+1)(2l+3)]
ll+1
ClearAll[h, f, Z, lmax, H, beta, Nmax, L1, L2, LPR,
p
To summarize, after casting the problem analytically
PR]
we use Mathematica programs to sequentially compute
lmax = 10;
Z(f) and P˜(f), then P(z), then S(r)=−2rdP(r)/dr =
h = .005;
4πr2Q˜(r) and finally Q˜(~r). We then consider the scaled
final = {};
variable ρ~ = ~r. Q(0) is then computed by considering
β For[k = 0, k <50, k++,
Q(ρ~) at ρ~ =~0 and plotting it as a function of β. Q0 = beta = .25*k + 1;
Q(0). BelowwedisplaytheprogramsforcomputingQ(ρ~) Nmax = Piecewise[90000,beta <=3, 9000, beta >3];
β3
andQ . We haveinsertedsome comments as partofthe L = {};
0
programs for clarity. For[n = 0, n<Nmax+1, n++, f = h*n*I;
H =Table[Switch[i−j,−1,f ∗(i+1)/Sqrt[
(2i+1)(2i+3)],0,i(i+1)/2,1,f∗(i)/Sqrt[(2i−1)(2i+
Program for computing Q(ρ~) 1)], 0],i,0,lmax,j,0,lmax];
,
M = MatrixExp[-beta*H];
ClearAll[h,f,Z,lmax,H,beta,L1,L2,LPR,PR] Z = M[[1, 1]];
lmax=10; L = Append[L, Z]]
Nmax=3000; L = Re[L];
beta=3; Pz = {};
h=.005; P1z = {};
L={}; For[l=−2,l<1200,l++,xi=.001∗l;
For[n=0,n<Nmax+1,n++, P = (h*beta/Pi)*Sum[L[[n]]*Cos[(n - 1)*h*xi*beta], n,
f=h*n*I; 1, Nmax];
H =Table[Switch[i−j,−1,f ∗(i+1)/Sqrt[ Pz = Append[Pz, xi, P];
(2i+1)(2i+3)],0,i(i+1)/2,1,f∗(i)/Sqrt[(2i−1)(2i+ P1z = Append[P1z, P]];
1)], 0],i,0,lmax,j,0,lmax]; V = P1z;
,
M=MatrixExp[-beta*H]; QR1 = {};
(*Computation of Z(f)*) L1 = Drop[V, 2];
5
L2 = Drop[V, -2]; [4] A. Ott, M. Magnasco, A. Simon, and A. Libch-
LPR = (L1 - L2)/(.001*2); aber, Phys. Rev. E 48, R1642 (1993), URL
LPR = Drop[LPR, 1]; http://link.aps.org/doi/10.1103/PhysRevE.48.R1642.
[5] T. Sanchez, I. M. Kulic, and Z. Dogic,
QR = Table[LPR[[i]] ∗ 1/((i − 1) ∗ .001) ∗ [−1/(2 ∗
Phys. Rev. Lett. 104, 098103 (2010), URL
Pi)],i,2,1199];
http://link.aps.org/doi/10.1103/PhysRevLett.104.098103.
QR1=Table[(i−1)∗.001,LPR[[i]]∗
[6] M.Mavrakis,T.F.-C.Azou-Gros,Yannick,J.Alvarado,
(1/(i−1))∗1/((i−1)∗.001)∗[−1/(2∗Pi)],i,2,2]; F.Bertin,AurelieIv,B.S.Kress,Alla,G.H.Koenderink,
final=Append[final,beta,(1/beta3)∗QR[[1]]]] and T. Lecuit, Nature Cell Biology 16, 322 (2014).
ListPlot[final] [7] J. Samuel and S. Sinha, Phys.
Rev. E 66, 050801 (2002), URL
http://link.aps.org/doi/10.1103/PhysRevE.66.050801.
[8] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts,
and P. Walter, Molecular Biology of the Cell (Garland
Science, 2008).
[9] J. Samuel and S. Sinha, Phys.
[1] C. Bustamante, S. B. Smith, J. Liphardt, and
Rev. Lett. 90, 098305 (2003), URL
D. Smith, Current Opinion in Structural Biol-
http://link.aps.org/doi/10.1103/PhysRevLett.90.098305.
ogy 10(3), 279 (2000), ISSN 0959-440X, URL
[10] A. J. Spakowitz, EPL (Europhysics
http://www.sciencedirect.com/science/article/pii/S0959440X00000853.
Letters) 73(5), 684 (2006), URL
[2] S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P.
http://stacks.iop.org/0295-5075/73/i=5/a=684.
Sethna, and M. D. Wang, Phys. Rev. Lett. 100(14),
[11] J. Shimada and H. Yamakawa, Macromolecules 17, 689
148301 (2008).
(1984).
[3] B.C.Daniels,S.Forth,M.Y.Sheinin,M.D.Wang,and
J. P. Sethna,Phys. Rev.E 80(4), 040901 (2009).