Polymerization of proteins actin and tubulin: the role of
nucleotides ATP, GTP
P. Ballone
Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany and
Dipartimento di Fisica, Università di Messina, I-98166 Messina, Italy.
J. Akola and R. O. Jones∗
Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany
(Dated: May 4, 2005)
Abstract
The polymerizing proteins actin and tubulin are found in almost all cells and are crucial to
important biological processes. Polymerization in both requires complexation by a nucleotide
(adenosine triphosphate (ATP) and guanosine triphosphate (GTP), respectively), whose role is
not understood in detail. Both reactions are entropy-driven, and we suggest that this arises from
the softening on polymerization of vibrational modes localized near ATP and GTP. Simulations
for a mesoscopic model based on particles and harmonic oscillators reproduce the transition from
a dilute, gas-like state at low T to filaments at high T and support this assignment.
1
Reversible polymerization of proteins is essential to cell motility and replication [1, 2]
and has motivated much study of two of the most important, actin and tubulin. The
monomers (Fig. 1) show important parallels: The actin monomer (G-actin, molecular weight
42 kDa, Fig. 1(a)) consists of four subdomains with a cleft containing a bound nucleotide
(adenosine triphosphate ATP or diphosphate ADP) and a divalent cation Mg2+ or Ca2+
[3, 4]. Polymerization to F-actin is followed by hydrolysis of the ATP molecule [4] to ADP
and an inorganic phosphate group Pi . The αβ-tubulin dimer (molecular weight 55 kDa, Fig.
1(b)) [5] is the basic unit of microtubules, which are essential components in all eukaryotic
cells and show dynamic instability that is based on binding and hydrolysis of a related
nucleotide (guanosine triphosphate GTP).
There have been many experimental studies of polymerization in actin and tubulin. The
measured T ∆S in actin polymerization exceeds the reaction enthalpy ∆H [6, 7], and entropy
as a driving force is consistent with the increase in polymerization rate in both actin and
tubulin solutions as temperature T increases [8, 9]. The hydrolysis rate of ATP increases
by more than three orders of magnitude when G-actin polymerizes [10], which has been
attributed to the weakening of ATP bonds by the F-actin environment. This leads to a
metastable ADP–Pi unit, whose decay is hampered by steric effects but ultimately destabilizes the polymer to give ADP-bound G-actin monomers [11, 12]. This mechanism can be
viewed as a “cytoskeletal timekeeper” that marks older filaments for depolymerization.
The source of entropy driving these transitions is not clear. The structure of the F-actin
subunit [13] differs little from that of G-actin [3], indicating that other factors are more
important. Contributions could come from domain motion in F-actin [14] or from water
osmosis from actin-ATP on polymerization [15]. We focus here on the dynamical properties
of actin-ATP, and we propose that entropy provided by the softening of ATP vibrational
modes offsets the loss of translational entropy and the increase in potential energy upon
aggregation. This picture is supported by simulations of a model that includes essential
features of the actin-ATP system and displays an equilibrium phase diagram similar to
those measured. A related mechanism should be active in tubulin.
Typical results for the polymerization extent Px (the percentage of monomers incorporated into aggregates) for G-actin solutions [9] are shown in the inset of Fig. 2. The rise
of Px at ∼ 300 K is a clear sign of entropy-driven polymerization. Full depolymerization
at low T implies that aggregation takes place against a potential energy contribution, rep2
resented by a repulsive Coulomb interaction between monomers [16]. Polymerization is
also opposed by translational (ideal) entropy, which decreases upon aggregation, so that
vibrational entropy remains as a likely driving force. Equilibrium polymerization driven by
entropy is not common, but it is well studied in inorganic (sulfur, phosphorus) and organic
(poly-(α-methylstyrene)) materials [17, 18].
In the (classical) harmonic approximation, the vibrational entropy of actin-ATP is:
S/KB = −
k
X
log[h̄ωi /KB T ],
(1)
i=1
where KB is the Boltzmann constant, and k is the number of vibrational modes with frequency ωi . If a single mode in G-actin changes frequency from ωG to ωF on aggregation,
the entropy change per monomer is:
∆S = SF − SG = KB log[ωG /ωF ].
(2)
Grouping monomers into N 0 = N/P polymers of length P raises translational entropy by
∆Sideal , depending on the thermodynamic conditions. If ∆U is the increase in internal energy
upon polymerization, and the monomer-monomer interaction is repulsive, polymerization is
possible above the floor temperature Tf = ∆U/[∆Sideal + ∆Svib ] whenever ∆Sideal + ∆Svib <
0.
The harmonic stretching frequency of the ADP–Pi covalent bond is predicted by density
functional methods [19] to be ωG = 924 cm−1 . ADP–Pi does not dissociate in the cleft of
G-actin, suggesting that the dramatic reduction of the ADP–Pi stretching frequency in Factin (ωF ∼ 0) could provide a free energy contribution at high T large enough to overcome
the ∆U and ∆Sideal contributions of opposite sign.
We have developed a model comprising spherical “particles” (representing the protein
monomers) that interact with purely repulsive forces and carry one or more internal variables
each (intramolecular vibrational modes), whose dynamics depend on the particle coordination. The model leads to ordering in an initially disordered configuration as T is increased,
and a suitable choice of functional form and parameters results in long, semiflexible polymers
reminiscent of actin and tubulin.
The model describes an assembly of N particles interacting by a pair potential, supplemented by bending contributions. A purely repulsive pair potential is adopted to emphasize
3
the relative roles of potential energy and entropy:
σ
1X
Upp ({RI }) =
2 I6=J |RI − RJ |
!α
fcut (|RI − RJ |),
(3)
where the cut-off function fcut is 1 (R < Rc ); 0 (R > Rc + ∆), and:
1
fcut (R) = {1 − cos [π(R − Rc )/∆]}; Rc ≤ R ≤ Rc + ∆.
2
(4)
Rc and ∆ are free parameters. The bending energy is given by:
Ubend = Kbend
N
X
[cos θjik − cos θ̄]2 ×
I6=J6=k
fcut (|RJ − RI |)fcut (|RK − RI |).
(5)
The choice cos θ̄ = −1 favors the linear structures familiar from actin microscopy [20].
Each particle carries s intramolecular degrees of freedom or hidden variables, for which
we use harmonic oscillators {ri,I , I = 1, ..., N ; i = 1, ..., s} with potential energy:
Ug ({r}) =
N
s
X
1X
KI [ri,I − RI ]2 .
2 I=1
i=1
(6)
The force constant KI depends on the coordination nI of particle I:
KI = K (0) + K (2) [nI − n̄]2 .
(7)
Here n̄ is a reference coordination, and nI is defined as:
nI =
X
fcut (|RI − RJ |).
(8)
J6=I
For convenience we use the same fcut as in Eq. 3.
Simulations have been performed for systems of 4000 atoms, using molecular dynamics
(MD) in the NVT ensemble and a 2D geometry to allow easy visualization of the results. No
qualitative changes are expected in 3D. The model parameters are: σ = 1, α = 6, Rc = 1.5,
δ = 0.5, and the parameters n̄ = 2, K (0) = 3 and K (2) = 6 imply that deviations from
the optimal coordination lead to a significant frequency increase and vibrational entropy
decrease for the corresponding oscillators. The choice n̄ = 2 implies that monomers prefer
chainlike structures, and s = 2 means that each particle I carries four independent harmonic
oscillators with the same frequency determined by KI and m. The simulation cell is a square
of side L, and simulations have been performed at densities ρσ 2 = N σ 2 /L2 = 0.05, 0.1, 0.125.
4
Repeated quenches and annealing cycles at low density (ρσ 2 = 0.1) led at T = 0 to a
disordered configuration of isolated particles with separations greater than Rc . Increasing
the kinetic energy results initially in nearly elastic collisions, followed by the separation
of the particles, and then to more persistent changes. Px (for ρσ 2 = 0.1, Fig. 2) shows
a monotonic rise with increasing T , reaching 80 % of the total mass at T = 0.6. If the
oscillator frequency is independent of particle coordination (see Fig. 2), Px (T ) does not
rise above 18 %, confirming that the aggregation displayed by the model is due to the free
energy gain provided by the oscillators, not to the accidental superposition of particles due
to thermal motion and non-vanishing density. The repulsive interaction means that the
average potential energy increases monotonically with increasing T .
Short linear units (dimers, trimers, etc.) predominate at T ∼ 0.1, with aggregates of 10-15
particles appearing occasionally. At intermediate (T ∼ 0.5) and high temperatures (T ≥ 1)
the dominant species are open chains of medium length (∼ 20 particles, Fig. 3), but larger
molecules (up to 80 particles) are also present. These structural changes are reflected in a
pronounced T -dependence of the radial distribution function (Fig. 4). The simulation results
depend strongly on density, with even slight increases above ρσ 2 = 0.1 greatly favoring
aggregation, which takes place at lower T and produces much larger molecules. On the
other hand, no visible aggregation takes place at the lowest density (ρσ 2 = 0.05) over the
temperature range we have explored (T ≤ 1.5).
The results at ρσ 2 = 0.1 do not reproduce the rapid, high-T depolymerization found
in actin solutions as T increases. Possible reasons for depolymerization are: (1) Increased
thermal motion reduces screening by counter-ions and strengthens the repulsive Coulomb
interaction between actin monomers; (2) the weakly bound Pγ phosphate groups escape
more easily from G-actin filaments as T increases, thus removing the primary source of
cohesion.
We have studied these effects by modifying the model: The decrease of screening with
increasing T can be incorporated by replacing the pair potential of Eq. (3) by a screened
Coulomb interaction VDH (r) = Q2 exp(−λDH r)/r with a Debye-Hückel screening length
(λDH = [KB T /4πQ2 ρ]1/2 ) that increases monotonically with increasing T . The density
and charge of the monomers are ρ and Q, respectively. The second effect is modeled by
attenuating the link between coordination and the KI whenever the elongation of the s
5
harmonic oscillators carried by particle I is larger than a cut off value Dc :
KI = K (0) + K (2)
v
u s
uX

× g t |r
i=1
i,I

− RI |2  × [nI − n̄]2 ,
(9)
where (cf Eq. 4) g(R) is 1 (R < Dc ), 0 (R > Dc + Qc ), and
1
g(R) = {1 − cos [π(R − Dc )/Qc ]}; Dc ≤ R ≤ Dc + Qc .
2
(10)
Large values of Dc (and Qc ) mean that the cut off mechanism acts only at high T and becomes
more significant as T increases. It also affects mainly optimally coordinated particles, whose
soft KI allows large amplitude oscillations, and both effects mimic the behavior of actin.
Results for Rc = 0.6 and Qc = 0.6 (Fig. 2) show a clear maximum in Px at T = 0.55.
Polymerization in tubulin is also enhanced by increasing T and density.
One non-
exchangeable molecule of GTP is bound to α-tubulin, and β-tubulin can hydrolyze its bound
GTP to GDP–Pi , release Pi , and exchange bound GDP for GTP. The microtubule is stabilized at the ends by caps of GTP-subunits, the loss of which can lead to rapid depolymerization [21]. The hydrolysis rate of GTP in microtubules is orders of magnitude higher
than in isolated tubulin monomers [10]. Collective dynamic effects in tubulin, however,
lead to the sudden collapse of microtubules [21], indicating that the description of tubulin
polymerization requires additional ingredients.
Our mesoscopic model focuses on the environment-dependence of intra-molecular vibrational modes, and the simulations show intriguing similarities to the behavior of proteins
polymerizing with increasing T . Although the model neglects the double-strand helical
structure of F-actin and its polar terminations, the results suggest that actin polymerization can arise from the softening of intramolecular vibrational modes caused by the docking
of actin monomers. ATP may be essential simply because the modes affected are localized
on this molecule and perhaps on neighboring groups in the actin cleft [4] and on water
molecules in the first hydration shell [15]. The determining step in the model is the ADP–Pi
bond weakening, and the loss of phosphate by F-actin, often identified with ATP hydrolysis,
is secondary.
Our picture could be tested by comparing the vibrational spectra of the actin-ATP
monomer and its filaments. This is a formidable challenge, because detailed structural
information is lacking. Second, the calculation of free energy differences is hampered by the
complexity of the system and the lack of reliable force fields for the crucial ADP–Pi bond.
6
Third, the effects of solvent and environment must be included. Nevertheless, the investigation of the actin-ATP vibrational properties is an interesting and exciting subject. The
most promising approach may be a density functional-based simulation for ATP and for the
closest protein groups [22], matched to a classical description of the remaining components.
J.A. thanks the Bundesministerium für Bildung und Wissenschaft, Bonn, for financial
support within the MaTech-Kompetenzzentrum “Werkstoffmodellierung” (03N6015). The
simulations were performed in the FZ Jülich on Athlon and Xeon computers provided in
part by the same program.
∗
Electronic mail: [email protected]
[1] L. Stryer, Biochemistry, Fourth Edition (Freeman, New York, 1995).
[2] See, for example, J. A. Theriot, T. J. Mitchison, L. G. Tilney, and D. A. Portnoy, Nature
357, 257 (1992).
[3] W. Kabsch, et al., Nature 347, (1990).
[4] S. Vorobiev, et al., Proc. Natl. Acad. Sci. USA 100, 5760 (2003).
[5] R. B. G. Ravelli, et al., Nature 428, 198 (2004).
[6] M. Kasai, Biochim. Biophys. Acta 180, 399 (1969).
[7] H. J. Kinosian, L. A. Selden, J. E. Estes, and L. C. Gershman, Biochim. Biophys. Acta 11077,
151 (1991).
[8] D. K. Fygenson, E. Braun, and A. Libchaber, Phys. Rev. E 50, 1579 (1995).
[9] (a) P. S. Niranjan, et al. J. Chem. Phys. 114, 10573 (2001); (b) ibid, 119, 4070 (2003).
[10] T. D. Pollard and A. G. Weeds, FEBS Lett. 170, 94 (1984).
[11] M.-F. Carlier, J. Biol. Chem. 266, 1 (1991).
[12] R. Melki, S. Fievez, and M.-F. Carlier, Biochemistry 35, 12038 (1996). The liberation rate of
Pi in actin (0.0026 s−1 ) is much lower than the ATP hydrolysis rate (0.035 s −1 ).
[13] K. C. Holmes, et al., Nature 347, 44 (1990); M. Lorenz, D. Popp, and K. C. Holmes, J. Mol.
Biol. 234, 826 (1993).
[14] R. Page, U. Lindberg, and C. E. Schutt, J. Mol. Biol. 280, 463 (1998).
[15] N. Fuller and R. P. Rand, Biophys. J. 76, 3261 (1999).
[16] F. Oosawa, J. Polym. Sci. 26, 29 (1957).
7
[17] S. C. Greer, Annu. Rev. Phys. Chem. 53, 173 (2002).
[18] See, for example, P. Ballone and R. O. Jones, 119, 8704 (2003).
[19] J. Akola and R. O. Jones, unpublished.
[20] I. Fujiwara, et al., Nature Cell Biology 4, 666 (2002).
[21] H. Flyvbjerg, T. E. Holy, and S. Leibler, Phys. Rev. E 54, 5538 (1996).
[22] See J. Akola and R. O. Jones, J. Phys. Chem. B 107, 11774 (2003) for a density functional
study of ATP hydrolysis.
Figures
(a)
(b)
FIG. 1: Monomers of (a) actin (with ATP, after Ref. 4) and (b) tubulin (with GTP and GDP,
after Ref. 5).
8
100
80
75
Px[ Mass % ]
Px [ Mass % ]
100
60
40
50
25
20
0
10
20
30
40
T [C]
0
0.0
0.3
0.6
0.9
1.2
1.5
T
FIG. 2: Simulation results in 2D (ρσ 2 = 0.1). Solid dots: basic model (see text). Open squares:
modified model (Eq. 9). Open diamonds: control system (K I = K (0) for all I). Inset: Polymerization degree of a 2.93 mg/ml actin solution in a 9.0 mmol/l KCl/H 2 O buffer (after Ref.
9).
T=0
T=2
FIG. 3: Snapshots from simulations of the basic model.
4
T=1.5
g (r )
3
2
T=0.07
1
0
0
1
2
3
R/σ
9
4
5
FIG. 4: Temperature dependence of the radial distribution function for the basic model.
10