j.a. tuszynski, s. portet and j.m. dixon: nonlinear assembly kinetics and mechanical properties of...

8/3/2019 J.A. Tuszynski, S. Portet and J.M. Dixon: Nonlinear assembly kinetics and mechanical properties of biopolymers

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Nonlinear Analysis ( )

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1

Nonlinear assembly kinetics and mechanicalproperties of biopolymers3

Jack Tuszynskia,,1, Stphanie Portetb,2, John DixoncaDepartment of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J15

bThe Samuel Lunenfeld Research Institute,Room 1060, Mount Sinai Hospital, 600 University Avenue,

Toronto, Ontario, Canada M5G 1X57cDepartment of Physics, University of Warwick, Coventry CV4 7AL, UK

Abstract9

This paper discusses the role of nonlinearities in the physical description of key biomolecules that

participate in crucial subcellular processes, namely actin, microtubules and ions crowding around11these filaments. The assembly kinetics of actin is that of a nonlinear process that is governed by

coupled nonlinear equations involving monomer concentration and filament number as the dynamical13variables. The dendritic growth of actin networks in cell motility phenomena is described by the

coupling of actin filaments to the protein Arp2/3. We then discuss how coupled differential equations15describing the interactions between ions in solution and the filament they surround can lead to solitonic

signal transmission. We also investigate the role of nonlinear dynamics in the formation of micro-17tubules. Space-flight laboratory experiments have shown that the self-organization of microtubules

is sensitive to gravitational conditions. We propose a model of self-organization of microtubules in19a gravitational field based on the dominant chemical kinetics. The pattern formation of microtubule

concentration is obtained in terms of a moving kink. Finally, we present a model of elastic proper-21ties of microtubules describing a microtubule as an elastic rod. We found that when the microtubule

is subjected to bending forces, the tangent angle satisfies a Sine-Gordon equation whose solutions23

Corresponding author. Fax: +1 780 492 0714.E-mail address: [email protected] (J. Tuszynski).1This work was supported by grants from MITACS and NSERC awarded to J.A.T.2S.P. acknowledges the support of the Bhatia post-doctoral fellowship.

0362-546X/$ - see front matter 2005 Published by Elsevier Ltd.

doi:10.1016/j.na.2005.01.089
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describe kink and anti-kink bending modes that may propagate at a range of velocities along the length1of the microtubule.

2005 Published by Elsevier Ltd.3

Keywords:; ;

1. Introduction5

The cells cytoskeleton is composed of three different types of filaments organized in

networks: microfilaments (MFs), intermediate filaments (IFs) and microtubules (MTs).7

MFs are a double helix of globular actin subunits. In a lamellipod, the MF network appears

as a 3D gel which is involved in cell motility. The 2D arrangement of MFs in contractile9

fibers appears to form cable-like structures involved in the maintenance of the cell shape and

transduction pathways. The polymerization dynamics and filament organization of actin has11

been modeled by Edelstein-Keshet et al. [6,9]. In the configuration of parallel strands, MFs

often form the core of microvilli, while in an anti-parallel arrangement, actin in conjunction13

with myosin brings about muscle contraction in the presence of ATP. MFs are often found

with the lattice configuration near the leading edge of growing or motile cells where they15

provide greater stability to the newly formed region. New actin filaments are nucleated at

the leading edge of the cells growth and trailing MFs are disassembled [2,3].17

MTs are long, hollow cylindrical objects made up of typically 13 longitudinal protofila-

ments consisting of-tubulin heterodimers, and are involved in a number of functions of19 the cell such as cell shape maintenance, mitosis and intracellular transport. By biological

standards, MTs are rigid polymers with a large persistence length of 6 mm. From Janmeys21

experiments [14], MTs suffer a larger strain for a small stress compared to either MFs or

IFs. The rupture stress for MTs is very small and typically is about 0.40.5 N/m2 [14]. The23

ratio of flexural rigidity per unit length to thermal energy in the case of MTs is of the order

of 100, whereas for actin it is only 2, meaning that the MTs are not greatly influenced by25

thermal agitation [30].

2. Actin filaments27

In this section, we give examples of the strong dependence of actins dynamic behaviors

on nonlinearities: two models of actin polymerization and one model of transport induced29

by nonlinear properties of actin.

2.1. Polymerization dynamics of actin filaments31

Polymerization of F-actin from G-actin is a largely monotonic process that is dependent

on ATP. F-actin assembles according to a standard nucleationelongation mechanism. Once33

assembled, MFs have a diameter of about 8 nm. Oosawa and Asakura [22] established thatspontaneous polymerization of actin monomers requires an unfavorable nucleation step35

followed by rapid elongation. They also determined that actin filaments can break and anneal
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J. Tuszynski et al. / Nonlinear Analysis ( ) 3

end to end. Inclusion of a fragmentation reaction improved the fit of nucleationelongation1

mechanisms to the observed time course of polymerization [4,7,29]. Earlier models for actin

polymerization showed that the critical size for the nucleus is somewhere between a dimer3

and a trimer, and that the number of explicit nucleation steps does not affect the results ofthe model [7,11,12,26]. With these considerations in mind, a simple five-step model was5

proposed [28]:

A + Ak+1k1

A2 k+1 = 10M1 s1, k1 = 106 s1, (1)7

A + A2k+2k2

A3 k+2 = 10M1 s1, k2 = 103 s1, (2)

A + A3k

+3

k3

A4 k+3 = 10M1 s1, k3 = 10 s1, (3)9

A + A4k+4k4

N k+4 = 10M1 s1, k4 = 0 s1, (4)

A + Nk+k

N k+ = 10M1 s1, k = 1 s1, (5)11

where A, Ai , and N represent the concentrations of actin monomers, filaments of i actin

monomers, and all longer filaments, respectively. The rate constants for the last reaction13

have been experimentally measured [23] and lead to the correct critical concentration(Cc > 0.1M), but the other rate constants are only approximations from kinetic simu-15

lations, chosen to reproduce the time course of polymerization over a limited range of

actin monomer concentrations. Such polymerization curves have already been published17

[4,7,11,12,26,29]. Filaments longer than 4 subunits are assumed to be stable and the back-

reaction rate k4 is set to zero. The coupled first-order differential equations that arise from19the set of reactions above are stiff-equations due to the large differences in the forward

and back reaction rates. This set of equations produces correct polymerization curves, but21

the average length as a function of actin concentration is completely incorrect. The mean

lengths of the observed filaments are almost independent of the initial concentration of actin23

monomers, while this simple model predicts a mean length with quite a different behavior,especially at high concentrations of actin. To solve this problem, additional processes are25

added to the model, namely the reaction

N + N kakf

N, (6)27

where ka is the annealing rate and kf the fragmentation rate of a filament. Since two filaments

are joined to form a new filament, the annealing rate has a quadratic dependence29

N = k+AA4 kaN2 + kfN. (7)It is reasonable to assume that annealing of two filaments is limited by diffusion, since31filaments diffuse more slowly than monomers. Erickson [10] estimated the fragmentation

rate to be in the neighborhood ofkf 108 s1.33
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The amount of polymerized protein in the system is given by the expression1

P

=A0

A

2A2

3A3

4A4, (8)

where A0 represents the initial actin concentration. The average length of a filament is given3

simply by L = P /N and we get

N = k+AA4 kaN3

L+ kf1 P + kf2 P2, (9)5

where all of the constants are absorbed in the variable ka. The values for the rate constantsdepend on the actin concentration and the filament density and length.7

When actin is purified from cells some actin-associated proteins remain in the sample,

including the capping protein CapZ. A reaction for capping filaments can be written as9

CapZ + Nkz+kz

N, (10)

where N represents a filament that is capped at the barbed end preventing monomer11addition or dissociation as well as annealing. The constants for the above reaction are

kz+ 3.5M1 s1 and kz 3 104 s1 [25]. The total number of filaments in the13system is N + N and the average length is now given by

L=

P

N + N. (11)

15

A capped filament can still fragment, but it can only anneal with an uncapped filament,

since this requires at least one free barbed end. Since two uncapped filaments can anneal in17

two ways, but capped and uncapped filaments have only one method of annealing, the rate

of annealing is half the original value.19

2.2. A simplified model for actin dendritic network aggregation

The nucleation of new F-actin filaments is a prerequisite for cell shape remodeling and21

motility processes involving rapid actin polymerization. It can either involve a de novonucleation of new filaments or a catalytic nucleation from pre-existing filaments. In the23

de novo nucleation, a trimeric nucleation core initiates a new filament. New filaments

resulting from a catalytic nucleation can be initiated either by the fragmentation of pre-25

existing filaments or by a branching process requiring the presence of the Arp2/3 protein

complex which is bound to the barbed end or to the side of a mother filament with the new27

daughter filament polymerizing out from it at a 70 angle. The Arp2/3 complex has to beassociated with other molecules as WASP family proteins or cortactin to be active. During29

these rapid actin polymerization processes, other important mechanisms take place, such

as the capping of the branched ends by, for example, gelsolin, avoiding polymerization31

at these ends to not exhaust the G-actin pool and to promote polymerization at preciselylocalized sites. All these processes induce strong nonlinearities. The events considered in33

our simplified model follow.


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2.2.1. De novo nucleation1

The nucleation core of F-actin is a trimer. This type of nucleation is G-actin dependent.

3G+

F, (12)3

where G is a G-actin monomer and F is a free F-actin filament.

2.2.2. Catalytic nucleation (branching)5

The Arp2/3 complex is able to form new barbed ends by branching at the ends of existing

filaments. WASP (and/or cortactin) is required for activation of the Arp2/3 complex: it7

is a nucleation-promoting factor. This type of nucleation is F-actin dependent. Thus, the

catalytic nucleation involves different events. The first step is the activation of the Arp2/39

complex,

W + A + G Y, (13)11where W is a WASP molecule, A is an inactive Arp2/3 complex and Y is an active Arp2/3

complex. The second step of the catalytic nucleation is the branching process13

F + Y 2FN + W or FN + Y 2FN + W, (14)

where FN represents an F-actin filament involved in the dendritic network.15

2.2.3. Polymerization/depolymerization

F + G +

F or FN + G+

FN. (15)17

These equations represent the elongation/disassembly processes of filaments.

2.2.4. Debranching19

FN F + A. (16)

The turnover between ATP G-actin and ADP G-actin is assumed to be instantaneous. The21

sequestration of the soluble pool, the severing and the capping/uncapping of filaments are

not considered in the model: all these processes involve the action of other types of proteins,23for example gelsolin. Thus, the proposed model could describe the self-aggregation of F-

actin filaments into dendritic networks. The reversible processes of annealing/fragmentation25

are not considered in our model.

We assume that the inactive Arp2/3 complex concentration, A, and the WASP protein27

concentration, W, are constant. Thus, our model is composed of 4 state variables ( G, Y, F,

and FN) and 7 parameters (+, , , , +, , and ).29

G = 3F

Elimination 3+G3

Nucleation+G(F + FN)

Polymerization+ (F + FN)

Depolymerization G

Activation, (17)

Y = GActivation

Y (F + FN) Branching

, (18)

31


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0

1

00.05

0.10.15

0.20.25

0.30.35

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

GF

FN

0

1

0

0.1

0.2

0.3

0.4

0.8

0.6

0.4

0.8

0.6

0.4

0.20.2

0

0.05

0.1

0.15

0.2

GF

FN

Fig. 1. Insensitivity of the actin debranching process to the initial conditions and for various parameter choices

( = 5, + = 100 left; = 0.05, + = 10 right).

F = +G3Nucleation

FElimination

+ FNDebranching

F YBranching

, (19)

1

FN = 2F Y Branching

+ FNY Branching

FNDebranching

, (20)

where +

is the rate constant of filament nucleation by trimerization and

is the rate3

constant of nucleation core elimination. Here + and represent the rate constants ofassembly of a G-actin monomer to an F-actin filament and of disassembly of a G-actin5

monomer from an F-actin filament, respectively. Here is the rate constant for the Arp2/3

complex activation by WASP proteins. is the rate constant of the filament branching, and7

is the rate of the dissociation of a filament from the network, the rate of debranching. All

the rate constants and the initial concentrations are positive. (See Fig. 1 for an illustration9

of the dynamics of these equations.)

2.3. Actin filaments as nonlinear ionic LRC transmission lines11

In this section, we set up an electrical model of the actin filament using inductive, ca-

pacitive and resistive elements. We apply Kirchhoffs laws to that section of the effective13

electrical circuit for one monomer, M, which involves coupling to neighboring monomers.

We expect a potential difference between one end of the monomer generated between the15

filament core and the ions lying along the filament at one Bjerrum length away. Due to

viscosity we also anticipate a resistive component to these currents which we insert in se-17

ries with L and denote it by R1. In parallel to these components, there exists a resistance,

R2, acting between the Bjerrum ions and the surface of the filament. In series with this19

resistance, we have a capacitance, C0. We assume that the charge on this capacitor varies

in a nonlinear way with voltage. Thus for the nth monomer,21

Qn = C0(Vn bV2n), (21)


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where b is expected to be small. From Kirchhoffs laws,1

vn

vn+1 =

LdIn

dt +I

nR

1, (22)

where vn and vn+1 are the voltages across neighboring resistors. Similarly, if the voltage3across the capacitor is Vn + V0, where V0 is the bias voltage of the capacitor,

vn = R2(In1 In) + V0 + Vn. (23)5

Furthermore, the current difference is In1 In = dQn/dt. Combining and transformingthese equations, we obtain7

L

d2Qn

dt2 = Vn+1 + Vn1 2Vn R1C0d

dt (Vn bV2n)

R2C0

2d

dt(Vn bV2n)

d

dt(Vn+1 bV2n+1)

ddt

(Vn1 bV2n1)

. (24)

A continuum approximation using a Taylor expansion in a small spatial parameter a9

gives

LC0j2V

jt2 = a2(jxx V )+R2C0j

jt (a2(jxx V ))R1C0

jV

jt +R1C0 2bVjV

jt , (25)11

which forms the basis of our physical analysis of the ion conduction problem [27].

As a result of applying an input voltage pulse with an amplitude of approximately 200 mV13

andadurationof800 s to an actin filament, electrical signals measured at the opposite end of

the actin filament reached a peak value of approximately 13 nA and lasted for approximately15

500s [17]. In a related earlier experiment [18], the wave patterns observed in electrically

stimulated single actin filaments were remarkably similar to recorded solitary waveforms17

from various experimental studies on electrically stimulated nonlinear transmission lines

[16,19,21].19

Solutions of Eq. (25) are taken as traveling waves, so the voltage is a function of a

moving coordinate, = x v0t, where v0 is the propagation velocity. Eq. (25) then21becomes

R2a2v0

L

d3V

d3+ (v20 c20a2)

d2V

d 2+

2bR1v0

LV R1v0

L

dV

d= 0, (26)

23

where c20 = 1/(LC0). Eq. (26) may be integrated once to yield

R2a2v0

L

d2V

d2 + (v20 c20a2)dV

d +1

22bR1v0

L V R1v0

L2 L

2bR1v0 = d0, (27)25

where d0 is an integration constant.
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To find solutions in terms of Jacobi elliptic excitations, we choose the propagation ve-1

locity, v0, so that v20 = c20a2 = v2max. For convenience, we define a shifted voltage by

W

=V

1/(2b) and obtain an analytical solution in the form3

W = w3 (w3 w2)sn2

3a2R2(w3 w1)2

2bR1( 0), k

, (28)

where 0k1 with k = (w3 w2)/(w3 w1) where k is the elliptic modulus. When5k 0 the sn() function tends to the trigonometric sine function. Simultaneously, theamplitude of the wave vanishes and so does the wavelength. On the other hand, when7

k 1 we obtain

W=

w1 +

2(w2 w1)1 + cosh(w2 w1( 0))

. (29)9

This solution represents a localized bump propagating with velocity v0. The waves in Eq.

(28) and the solitary wave in Eq. (29) travel at the velocity v0 estimated as vmax=3105 m/s,11which is astonishingly high indicating a purely electromagnetic resonant energy transfer

from one LC element to the neighbor with no loss.13

The voltage equation, Eq. (27), may be cast into the Fisher form by denoting d V /d as

V, and rewriting it as V + V + V2 + V + = 0 where15

=

(v20 c20a2)L

R2a2

v0

,

=

bR1

a2

R2

,

=

R1R2a

2, and

=

R1

R2

1

4ba2

d0L

R2a2

v0

. (30)

We define a change of variable V = W + , where and are chosen as17

=2 4

and = +

2 4

2. (31)

Hence the traveling wave solution found is19

V =1

2b + d0LbR1v0 1 21 + p expq2

a2R2 bd0R1Lv0 s

. (32)The propagation velocity for the traveling kink solution is21

v = D(v20 v2max)LR2a2v0

. (33)

The velocity of ion flows in physiological situations, such as action-potential propagation23

[1], ranges between 0.1 and 10 m/s; hence we estimate the corresponding nonlinear wave

velocities,

v, to be 1 m/s

v100m/s. Actin interacts with a number of ion channels, of25

different ionic permeability and conductance. Thus, it is expected that channel opening,single-channel currents and other channel properties, including the resting potential of the27

cell, may significantly modify the amplitude and velocity of the soliton-supported waves.


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3. Microtubules1

In this section, we give two examples of microtubule dynamics that are dominated by

nonlinearity.3

3.1. MT polymerization in reduced gravitational fields

Space-flight laboratory experiments have shown that the in vitro and in vivo self-5

organization of MTs is sensitive to gravitational conditions. In a previous work, we have

proposed a model of self-organization of MTs in a gravitational field [24]. This reaction7

diffusion model was based on the dominant chemical kinetics and on the interplay between

the diffusion processes and the drift induced by gravitational fields. We have considered as9

chemical kinetics, the nucleation of filaments and the elimination of the nucleation center,the elongation and the shrinking of filaments, the recycling of GDP-tubulin in GTP-tubulin.11

It has been demonstrated that the polymerization of MTs proceeds in a traveling wave

fashion propagating with a constant velocity v. The tubulin concentration, C, as a function13

of the moving coordinate, = x vt, describes a polymerization wave as follows:

C = 22/n

1 tanh

n

2

2n + 4 ( 0)2/n

, (34)15

where 0 is an arbitrary constant and n represents the number of dimers involved in the

nucleation center. The wave velocityv

is dependent onn

,v

= (n

+ 4)/(

2

n+ 4

)

Dc

k117 with Dc the diffusion coefficient of tubulin dimers, and k1 the recycling rate of GDP-tubulin

in GTP-tubulin. The nucleation process has been classically described as the power of the19

GTP-tubulin concentration with nucleation exponents ranging from 6 to 12. This term

represented the strongest nonlinearity of the model.21

3.2. The bending dynamics of MTs

In this section, we represent an MT as an elastic rod which can undergo large bending23

motions. We first define the arc length, s, belonging to [0, L], where L is the MT length,(s,t) is the tangent angle to the MT at position s and at time t. At time t, the curvature,251/Rc, of the MT at the position s is 1/Rc = j/js where Rc is the radius of curvature. TheLagrangian density can be expressed in terms of as follows [5]:27

L= I2L

j

jt

2f

2

j

js

2 F cos

, (35)

where I is the moment of inertia andf is the flexural rigidity [13]. The term (I /2L)(j/jt)229

represents the kinetic energy density, where j/jt is the angular velocity. The second term,

(f/2)(j/js)2, is the local elastic potential energy, and the third term,

F cos measures31

the energy density expended in applying a force, F, to the MT. The moment of inertia, I, ofa rod, representing an MT, around an axis perpendicular to its length and passing through33

one of its ends, is given by I = mL2/3, where m is the total mass of an MT of length L.


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The dynamics of the bending of an MT resulting from the EulerLagrange equation is1

expressed by the nonlinear wave equation

IL

j2jt2

+ f j2js2

F sin = 0 (36)3

where the propagation velocity of the bending wave is v=fL/I. The change of variables,x =

Ff

s and t =

F LI

t, rescales Eq. (36) to5

j2

jt2 j

2

jx2= sin , (37)

where the tildes are omitted for simplicity. Eq. (37) has the form of the unperturbed Sine-7

Gordon equation. It has one particularly important solution taking the form of a kink[15,20],

(x,t) = 4tan1exp

(x vt x0)

1 (v/vm)2

, (38)

9

where vm is the maximum phonon velocity. In addition, the Sine-Gordon equation [8] has

an infinity of multi-soliton solutions T, but their physical significance is unclear.11

4. Summary and conclusions

Nonlinearities are very important in the physical description of several key biomolecules13

that participate in a crucial subcellular processes, namely actin, MTs and motor proteins (es-

pecially kinesin). We showed in this paper that the assembly kinetics of actin is a nonlinear15

process that requires not only a mechanism of saturation but also annealing and fragmen-

tation that are governed by coupled nonlinear equations involving monomer concentration17

and filament number as the key variables. The observed dendritic growth of actin networks

in cell motility phenomena is subsequently described by the coupling of actin filaments to19

the protein calledArp2/3. Next, we discussed how coupled differential equations describing

the interactions between ions in solution and ions on the filament can lead to solitonic signal21transmission.

We then discussed the role of nonlinear dynamics in the formation of MTs. First of all,23

it has been known for more than a decade that high concentration of the constituent protein

tubulin leads to self-sustained oscillations in the assembly and disassembly dynamics of25

MTs. We proposed a model of self-organization of MTs in a gravitational field. The model

is based on the dominant chemical kinetics. The pattern formation of MT concentration is27

obtained (1) in terms of a moving kink in the limit when the disassembly rate is negligible,

and (2) for the case of no free tubulin and only assembled MTs present [24]. The results of29

our simulations are in good quantitative agreement with experimental data.

We presented a continuous medium model of the elastic properties of MTs that includes31large bending motions of MT filaments by describing a MT as an elastic rod. We found

that when a MT is subjected to bending forces, the tangent angle satisfies a Sine-Gordon33
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equation. Particular analytical solutions of this equation describe kink and anti-kink bending1

modes that may propagate at a range of velocities along the length of the MT.

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j.a. tuszynski, s. portet and j.m. dixon: nonlinear assembly kinetics and mechanical properties of...

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