Constriction model of actomyosin ring for cytokinesis by fission yeast using a twostate sliding filament mechanism Yong-Woon Jung and Michael Mascagni Citation: The Journal of Chemical Physics 141, 125101 (2014); doi: 10.1063/1.4896164 View online: http://dx.doi.org/10.1063/1.4896164 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Investigation of the effects of cell model and subcellular location of gold nanoparticles on nuclear dose enhancement factors using Monte Carlo simulation Med. Phys. 40, 114101 (2013); 10.1118/1.4823787 Monte Carlo investigation of the increased radiation deposition due to gold nanoparticles using kilovoltage and megavoltage photons in a 3D randomized cell model Med. Phys. 40, 071710 (2013); 10.1118/1.4808150 BioPhotonics workstation: A versatile setup for simultaneous optical manipulation, heat stress, and intracellular pH measurements of a live yeast cell Rev. Sci. Instrum. 82, 083707 (2011); 10.1063/1.3625274 On the relationships between kinetic schemes and two-state single molecule trajectories J. Chem. Phys. 123, 064903 (2005); 10.1063/1.1979489 Mathematical model of the cell division cycle of fission yeast Chaos 11, 277 (2001); 10.1063/1.1345725

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THE JOURNAL OF CHEMICAL PHYSICS 141, 125101 (2014)

Constriction model of actomyosin ring for cytokinesis by fission yeast using a two-state sliding filament mechanism Yong-Woon Jung1 and Michael Mascagni2,a) 1

Department of Chemistry, Texas A&M University, College Station, Texas 77843-3255, USA Departments of Computer Science, Mathematics and Scientific Computing, and Graduate Program in Molecular Biophysics, Florida State University, Tallahassee, Florida 32306-4530, USA 2

(Received 2 May 2014; accepted 8 September 2014; published online 29 September 2014) We developed a model describing the structure and contractile mechanism of the actomyosin ring in fission yeast, Schizosaccharomyces pombe. The proposed ring includes actin, myosin, and α-actinin, and is organized into a structure similar to that of muscle sarcomeres. This structure justifies the use of the sliding-filament mechanism developed by Huxley and Hill, but it is probably less organized relative to that of muscle sarcomeres. Ring contraction tension was generated via the same fundamental mechanism used to generate muscle tension, but some physicochemical parameters were adjusted to be consistent with the proposed ring structure. Simulations allowed an estimate of ring constriction tension that reproduced the observed ring constriction velocity using a physiologically possible, selfconsistent set of parameters. Proposed molecular-level properties responsible for the thousand-fold slower constriction velocity of the ring relative to that of muscle sarcomeres include fewer myosin molecules involved, a less organized contractile configuration, a low α-actinin concentration, and a high resistance membrane tension. Ring constriction velocity is demonstrated as an exponential function of time despite a near linear appearance. We proposed a hypothesis to explain why excess myosin heads inhibit constriction velocity rather than enhance it. The model revealed how myosin concentration and elastic resistance tension are balanced during cytokinesis in S. pombe. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4896164] NOMENCLATURE

ERT MCE RF SL SSL T-V

elastic resistance tension minimal contractile element reduction factor sarcomere-like semi-sarcomere-like tension-velocity

I. INTRODUCTION

During eukaryotic cytokinesis, a ring-like structure composed predominantly of actin and myosin filaments assembles on the inner surface of the plasma membrane at the future site of cell division.1–6 Marsland and Landau hypothesized the existence of the contractile ring1 and the filamentous structure of the ring was first observed by Schroeder in electron micrographs of the jellyfish Stomatoca atra eggs.2 The existing hypothesis of an actomyosin ring using ATP like primitive muscle was suggested more logically by Norman Wessells group3 and was tested by Schroeder.7 Schroeder verified the hypothesis, having observed that the actin-like microfilaments of Hela cells bind heavy meromyosin, the soluble tryptic fragment of muscle myosin, from rabbit skeletal muscle.7 The actomyosin ring pulls the membrane inward as it constricts, ultimately forming two daughter cells. In fission yeast and other cells with walls, this also guides the formation of a) Author to whom correspondence should be addressed. Electronic mail:

[email protected] 0021-9606/2014/141(12)/125101/14/$30.00

the cell-wall septum.6 Other proteins are recruited to the ring during various stages of assembly and maturation,8 including α-actinin, actin depolymerization factor (ADF), and formin, an actin polymerization protein.4 Actin polymerizes and the resulting chains self-associate to form double-stranded semi-flexible directional filaments with barbed (+) and pointed (−) ends. Barbed ends may be attached to the cytoplasmic surface, possibly by formins and other anchor candidate proteins,9 such that the plasma membrane pulls inward when actin and myosin filaments interact.10 Actin monomers bind to existing filaments ca. 10times faster at barbed ends than at pointed ends, resulting in a non-steady-state tread-milling movement in which barbed ends lengthen as pointed ends shorten.11, 12 ATP-bound actin monomers add to barbed ends. These units hydrolyze in time, forming a gradient within filaments, with ADP-bound units dominating at pointed ends and ATP-bound units at barbed ends [Fig. 1(d)]. Myosin-II, referred to here as myosin, is a two-headed motor protein that polymerizes to form bipolar filaments. Myosin and actin filaments interact to generate the force required for contraction.10, 13–15 During ring assembly, myosin filaments intercalate in parallel fashion between actin filaments, eventually forming an actomyosin ring that is aligned circumferentially around the cell’s equator.2, 14–18 In the presence of ATP, myosin heads ratchet along actin filaments towards their barbed ends as the heads attach and detach. Once at barbed ends, the heads presumably attach randomly to other actin filaments, and the process continues until ring constriction is complete.

141, 125101-1

© 2014 AIP Publishing LLC

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J. Chem. Phys. 141, 125101 (2014)

FIG. 1. Proposed structure of the actomyosin ring. (a) assembly of SL (left) and SSL (right) MCEs from nodes. The MCE structure consists of a myosin bipolar filament (blue in center), actin filaments (red), α-actinin dimers (dumbbell-shape), and formin monomers/dimers (filled rectangles). ((b) left) a torus ring composed of 16 half-SSL units linked end-to-end. ((b), right)) a cross-section of the ring in a region containing myosin filaments (blue), myosin head groups (green sticks with black circles) interacting with actin filaments (red). Each internal actin filament may interact with myosin head-groups from three surrounding myosin filaments; each internal myosin filament may interact with six surrounding actin filaments in a hexagonal lattice. (c) a well arranged (hypothetical) SL (left) and SSL (right) ring structure. SL ring structure shows that all the pairs of actin:myosin interactions are viable and positioned with the exact alignment to maximize the contraction force. We first calculated this muscle sarcomere-like (SL) tension before converting it to (SSL) ring constriction tension. SSL ring structure (randomly arranged, but assumed realistic) shows that due to stereochemical differences only some of all the possible actin:myosin interactions will lead to an effective contraction when actin filaments span both ends of a myosin biopolar filament. (d) the disassembly of actin filaments exerted by the ADF activity. T, D-p, and D denote each actin subunit with ATP, ADP and phosphate, and ADP, respectively.

α-actinin crosslinks actin filaments during ring assembly; this appears to maximize the contractile force during maturation.19 α-actinin dimers enforce the parallel orientation of actin filaments by binding to adjacent filaments, thereby creating a constant spacing into which myosin polymers intercalate. Because contraction velocity depends on the distance between parallel actin and myosin filaments, and the degree to which they are aligned,20 the extent of alignment appears to be related to the concentration of α-actinin.21 The

length of α-actinin is evolutionally conserved among various organisms,22 suggesting a consistent ring structure. Actin, myosin, and α-actinin are also found in muscle sarcomeres, the minimal contractile unit of muscle myofibrils.10, 13, 14, 21, 23 Actin in fission yeast, S. pombe, is ∼ 90% identical to mammalian muscle actin in terms of amino acid sequence; the two proteins are also structurally similar.24 S. pombe myosin is probably similar to muscle myosin in shape and size25 and it likely operates with the same ATPase mechanism.26 The

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general similarities of both proteins suggest that the mechanism of contraction is fundamentally the same for all actomyosin networks regardless of organism.10, 13, 14, 21, 27, 28 One critical difference between ring and muscle systems is that the ring dissolves during constriction, whereas muscle sarcomeres do not.28 This is likely due to the considerable activity of ADF, an enzyme that appears to be more active in the actomyosin ring than in muscle.29 Formins and other proteins that cap the barbed ends of actin are possibly involved in this ring disassembly.19, 30 ADF binds and destabilizes ADP-bound actin subunits at the pointed ends of actin filaments, accelerating the disassembly of terminal subunits.12, 31–33 This results in a faster dynamic turnover of actin subunits within filaments in the ring, relative to the turnover rate in muscle sarcomeres,29 with ring-filament rates corresponding to a half-life of < 1 min.4 In the presence of end-tracking cross-linkers, such as α-actinin, the tension generated by this turnover may augment the sliding filament tension to constrict the ring more rapidly.34–36 The mechanism of muscle contraction has been studied extensively, and many molecular level models have been developed. Huxley,37 Hill,38, 39 and others40–44 have made major contributions. Continuum-based models of ring constriction have also been proposed.35, 45–47 All molecular-level models of muscle contraction are based on the sliding filament mechanism of Huxley.37 In this mechanism, filament sliding is generated by interactions of myosin heads with actin filaments.44 Huxley assumed two states, one in which myosin heads are attached to actin filaments and the other in which they are unattached. He calculated the contraction force by considering the probability of a myosin head being attached, as attachment is necessary for contraction. Although Huxley’s original model reproduced the empirical tension-velocity (T − V ) relationship reported by (A.V.) Hill, there have been numerous efforts to improve the model by expanding the number of states and/or reaction pathways.38–40, 42, 48–54 (T.L.) Hill thoroughly considered the thermodynamics of muscle contraction and proposed a selfconsistent two-state sliding filament model that we adopted for our ring model. This model gives accurate and computationally efficient solutions describing the transduction of ATP hydrolysis energy into mechanical energy (the ratcheting movement of myosin heads) during physiologically relevant time scales.40, 54 The mechanism of actomyosin ring constriction is probably similar to that of muscle.12–15, 20, 28 Besides being composed of the same essential proteins, microscopic images of the ring indicate a tightly packed filamentous belt2, 8, 14–18 with uniformly distributed actin filaments, suggesting a sarcomerelike (SL) structure.55, 56 This belt has been estimated to be composed of ≥3 SL units, with each actin filaments ∼1 μm long.8 Despite these similarities, the sliding filament mechanism has not been applied to the case of the actomyosin ring.57 This may be due to difficulties in establishing a relationship between the constriction ring tension and velocity, in determining the molecular-level structure within the ring, or in determining cell resistance, i.e., the force opposing cell constriction.58

J. Chem. Phys. 141, 125101 (2014)

In this study, the open question is how, and with what mechanisms, actomyosin ring in fission yeast constricts. To address this question, we propose a semi-sarcomere-like (SSL) ring structure in S. pombe and an associated theoretical model of ring constriction (exclusively seeking the meaning of experimental data published). Cytokinesis in S. pombe is similar to that of animal cells and is ideal for model development because its inventory of contractile ring proteins has been quantified.8, 59 We further propose that this ring structure obeys a relationship during constriction that is fundamentally the same as that developed by Hill60 and Huxley37 for muscle sarcomeres. Numerical simulations using the modified relationship estimate ring constriction tension and reproduce ring constriction velocity. The purposes of this paper are to: (1) present a theoretical model of actomyosin ring constriction based on the experimental data and the sliding filament theory that is able to similarly elucidate the animal cell cytokinesis; (2) propose possible scenarios of how actin and myosin filaments are assembled to be a SSL structure; (3) investigate a way to compromise the physiological differences between muscle contraction and actomyosin ring constriction; and (4) demonstrate a theoretical way of how ring constriction tension is estimated and how ring constriction velocity is reproduced. The layout of the paper is as follows. In Sec. II, the processes for establishing the SSL ring model using a self-consistent two-state sliding filament mechanism are described. The ring structure, the ring assembly, and the ring constriction are illustrated in detail. In Sec. III, after a plausible set of physicochemical parameters is determined, various simulations using the SSL ring model are performed and the results are analyzed. Some conclusions and discussion are presented in Sec. IV. The Appendixes describe ring contractility, ring curvature, and changes in protein concentrations.

II. THEORY AND METHODS A. Ring structure

The ring is modeled as a torus with 10 μm circumference, 0.17 μm thickness, and 0.24 μm3 (2.4 × 10−16 L) volume.8 These dimensions are similar to those determined experimentally, including a ring thickness that is 0.1–0.2 μm.55, 61, 62 The ring model is composed exclusively of actin, myosin, and α-actinin proteins, with stoichiometric concentrations in the ring of 460, 20, and 4 μM, respectively, similar to values obtained experimentally.59, 63 These dimensions and concentrations are consistent with the presence of 66 700, 2900, and 580 monomers of actin, myosin, and α-actinin in the assembled ring.8, 59, 63 Actin monomers in the ring are organized into ca. 300 filaments,8 each initially 0.6 μm long.14, 18, 28 The exact number of filaments depends somewhat on the size of S. pombe.18, 36 Adopting a simple continuum concept, filaments were assumed to have equal lengths, but to be in a dynamic turnover via tread-milling. In this respect, ring constriction was interpreted as the progressive shortening of each actin filament,30 such that the disappearance of one actin filament corresponded to the complete contraction of the ring.

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In the ring model, α-actinin monomers are organized into ca. 300 dimers.8 Myosin monomers are organized into ca. 150 bipolar filaments,8 each 0.23 μm long and composed of 20 monomers,10, 14, 19, 64 with 20 heads on each pole. This is shorter than mammalian muscle myosin bipolar filaments, but is comparable to the length of thin myosin filaments in Acanthamoeba.64 Whether bipolar filaments form prior to or after ring assembly in real cells is uncertain.65 We propose that the ring is composed of a network of minimal contractile elements (MCEs) [Fig. 1(a)] that are organized into a SSL ring structure, circumscribing the inner surface of the cell membrane [Fig. 1(b)]. We hypothesize that these structures are loosely organized at the molecular level relative to muscle sarcomeres, but that they nevertheless constrict by a sliding-filament mechanism. In the ring model, actin, myosin, and α-actinin oligomers are organized into ca. 150 MCEs, each composed of 1 myosin bipolar unit, 2 actin filaments, and on average, 2 α-actinin dimers. The initial overall length of MCEs was set at 1.3 μm, including a 0.1 μm interval between the two actin filaments. These filaments have their pointed ends facing each other, which is critical for contraction. We view individual MCEs as able to contract, defined as actin filaments sliding towards each other, but lacking sufficient power or organization to contract the membrane with which they are associated. This ability may require 8 SSL units end-to-end circumscribing the ring, with neighboring SSL units linked by α-actinin dimers [Fig. 1(c)]. One SSL unit slice of the proposed actomyosin ring [Figs. 1(b) and 1(c), right] has ca. 10 myosin bipolar filaments, 19 actin filaments, and 18 α-actinin dimers.8 The SSL units would not show periodic spatial patterns [Fig. 1(c), left] consistent with that observed.66

B. Ring assembly

The strength of actin:myosin binding interactions should depend on the orientation of the filaments. Both actin and myosin filaments are directional, and we suspect that myosin heads have a stereochemical preference in binding actin filaments. Binding interactions that could lead to contraction (called viable interactions) have the pointed ends of two actin filaments directed towards the center of the myosin bipolar filament [symbolized → ∩ ←]. In muscle, the actin:myosin interactions in all hexagonal directions around a myosin filament are viable. Nonviable orientations [← ∩ →] will have no or only weak interactions. In situations where a myosin bipolar filament straddles a single actin filament, half of the interactions would be viable, and half would be nonviable. The ring in S. pombe assembles from 65 membranebound cytokinesis nodes close to the equatorial plasma membrane of the cell.19, 65 These nodes are discrete protein clusters dispersed during the G2/M transition65 that move randomly in 2D along the inner surface of the plasma membrane. Whether the nodes are essential for ring formation is uncertain, but they at least appear to guide its proper location.67 During anaphase A, the nodes gradually coalesce and dissolve as the ring forms.19

J. Chem. Phys. 141, 125101 (2014)

Each node typically has 2–3 myosin (possibly bipolar) filaments, 4–6 actin filaments, and 2–3 formin dimers. This correlates roughly to a total of ca. 150 formin dimers.8 The formin dimers within each node anchor the barbed ends of actin filaments to the nodes and also radiate the actin filaments with pointed ends.4, 8, 9, 19, 57 Thus, each node radiates filaments and can receive them from other nodes.8 Connections between nodes might be bidirectional, in which case, two actin filaments with opposite polarities are connected or unidirectional. According to Pollard’s Search, Capture, Pull, and Release mechanism,19 after such actin filaments randomly bind myosin filaments in an adjacent node due to Brownian motion, the two nodes pull together incrementally and then release due to frequent breaks in actin filaments. This allows nodes to have maximum flexibility for their directional movement in forming the ring.5 We have considered two possible scenarios in which myosin motor activity58 might help form MCEs between adjacent nodes. In one scenario [Fig. 1(a), right], actin filaments radiating from a central node bind to myosin motors in two adjacent nodes, whereas actin filaments from those adjacent nodes are received by the central node. Mutual contraction pulls the three nodes together, leading to a SSL MCE structure once the nodes dissolve. In this structure, two actin filaments are oriented viably to allow contraction (→ ∩ ←), whereas two others are not viably oriented (← ∩ →). In the other scenario [Fig. 1(a), left], actin filaments from adjacent nodes are also received by a central node. In this case, nodes must differentiate into those that contain formins and radiate filaments, and those that contain myosin and receive them. Such differentiation could afford a SL MCE structure once the nodes dissolve. In the SL MCE structure, all actin filaments are oriented viably. Although this would produce higher ring tension, we suspect that the SSL MCE structure dominates because the mechanism of assembly is presumed more persuasive. Finally, SSL MCEs eventually consist of a SSL actomyosin ring when this process is repeated. Pre-existing actin filaments transported from other regions of S. pombe also interact with myosin filaments in the nodes4 and such filaments may form a significant part of the ring structure.68 Pre-existing filaments could give rise to a SLtype structure such that the actomyosin ring may be composed of both SL and SSL structural elements [Figs. 1(a) and 1(c), left and right]. Our ring assembly mechanism is consistent with a computational study by Carlsson in which 70% of the actin:myosin interactions that were initially randomly distributed in an actomyosin system developed contractility after being subjected to an energy function reflecting actin:myosin interactions.69 A similar reorganization and maturation may occur in the actomyosin network of S. pombe. Here, Brownian thermal motion70 may cause myosin filaments to translate along actin filaments and form overlapped MCE configurations. Nonviable interactions would be highly dynamic and viable interactions less so, such that once viable interactions form, they would freeze out due to their increased stability. Once ring organization exceeds the cell resistance energy, the

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FIG. 2. Schematic of the two-state sliding filament model. Only one head of a myosin head group and one pointed end of an actin filament are shown. The unattached state consists of two substates in which ATP is hydrolyzed (a), and the head group changes conformation (b). The attached state consists of three substates [(c)–(e)].

mature ring would have the capacity to constrict, when triggered by the cell. C. Ring constriction

Several processes have been suggested as being responsible for ring constriction, including the power stroke generated by actin:myosin sliding interactions, actin turnover (treadmilling), sufficient ADF activity (accelerated actin depolymerization), and septum formation. We will not consider septum formation here due to its high uncertainty. If the power stroke generates the force required for ring constriction, then the sliding-filament mechanism developed for muscle contraction should also be used to understand ring constriction. Fig. 2 shows the relationship between ATP hydrolysis and the power stroke based on the two-state sliding-filament mechanism. The two-state sliding-filament model is a simplified version of a five-state model in which multiple states have been combined. There are two (unattached and attached) states and each state involves millisecond-scale transitions between many substates.54 In the unattached state, ATP hydrolysis within a myosin head causes a conformational change. This repositions the myosin head along the filament by ∼7 nm. Then Brownian thermal motion causes the myosin head to attach to a random position on an actin filament [Fig. 2(c)]. In the attached state, the power stroke follows the dissociation of Pi and is associated with the release of ADP [Figs. 2(c) and 2(d)]. Conversion to the unattached state is triggered when ATP binds to a myosin head [Figs. 2(e)– 2(a)]. The population of these states alternates periodically as myosin ratchets along actin filaments towards the barbed ends of those filaments. The ATP cycle [displacement (d) = 36 nm with successive six power strokes] involves ∼5 other myosin heads acting on different actin filaments.71 However,

J. Chem. Phys. 141, 125101 (2014)

the total displacement associated with the ATP cycle for the actomyosin ring in S. pombe has not been determined. The displacements are probably irregular due to imperfect phasing of the MCEs, which may produce lower tension than muscle. The actin turnover rate and ADF activity may also be responsible for ring constriction. In the ring, a coupled turnover rate and sufficient ADF activity results in the accelerated depolymerization of actin filaments and may expedite ring constriction, adding the tension generated from end-tracking cross-linkers. Accelerating depolymerization has been hypothesized and reported to be the main mechanism driving actomyosin ring constriction in fission yeast47 and budding yeast.72 The Pinto hypothesis that actin depolymerization drives actomyosin ring contraction may be valid, but only when the constriction force is not required for cytokinesis, as might be the case for budding yeast.73 The ring in C. elegans embryos progressively disassembles without frequent turnover of its structural components,30 implying that depolymerization is not required for cytokinesis in these cells; the same situation may hold for S. pombe. Moreover, the accelerated depolymerization in S. pombe may generate insignificant ring constriction tension relative to actin:myosin sliding, as suggested by the minimal effects of a ADF (cofilin) mutation on ring constriction velocity.74 In contrast, rings from cells containing mutant myosin or cells containing reduced amounts of myosin cannot constrict properly.36, 58, 63 In our model, we resolve these apparent incongruencies by proposing that the power stroke and the ADF-accelerated actin depolymerization may be synchronized to maximize the constriction efficiency. Similar ideas have been suggested by Carlsson (2006) and Carvalho et al. (2009). Collectively, the ring constriction in S. pombe appears to be mainly caused by interaction forces between actin and myosin filaments, whereas the accelerated depolymerization may help clear the actin filaments over which the myosin heads have already slid or other purposes. Thus, in our current model, we ignored the effects of turnover rate and accelerated depolymerization, which presumably do not significantly alter the strength of ring contraction very much. Further experimental studies are required to resolve this issue. During ring constriction and disassembly, the number of actin and α-actinin molecules decreases continuously, whereas the corresponding concentrations of these proteins does not vary.8 The number of myosin monomers in the ring and the thickness of the ring also do not vary much during constriction.4 At terminal stages of this process, myosin filaments may appear to overlap when viewed along one dimension. Ring thickness is presumably maintained as long as the concentration of α-actinin does not vary, because this protein maintains the same interval spaces between actin filaments regardless of the length of those filaments. The final process of cell division, called abscission,75 arises from a different mechanism that is not considered here. D. Muscle sarcomere tension

We introduce mathematical descriptions of a selfconsistent two-state sliding filament mechanism.38, 39

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J. Chem. Phys. 141, 125101 (2014)

According to Huxley,37 muscle force per myosin head varies with the probability of the head being attached, with only one head of a two-headed myosin active at a time.52 In a muscle sarcomere, n(x, t) is the probability that the head at time t and displacement x away from an equilibrium position is attached to actin filaments.76 The maximum displacement, where a myosin head binds to the actin filament at the positive side of the equilibrium position, is given by h. For any x between 0 and h(0 ≤ x ≤ h) (Fig. 2), the myosin head exerts a shortening (positive) force. For x ≤ 0 the attached myosin head exerts a lengthening (negative) force. For x > h no myosin heads are dragged into this region during the shortening.76 To obtain n(x, t)(0 ≤ n ≤ 1), Hill et al. (1975) developed a Huxley-type partial differential equation for the conservative reaction scheme involving the probability of a myosin head being attached for x ≤ h, ∂n(x, t) ∂n(x, t) dx dn(x, t) = + dt ∂t ∂x dt = [ff (x) + gr (x)][1 − n(x, t)] − [fr (x) + gf (x)]n(x, t).

(1)

Reversible attachment and detachment reactions are assumed. In one reaction, myosin bound with ADP and Pi attaches to actin filaments in accordance with forward and reverse rateconstants ff and fr . In the other reaction, myosin detaches from actin after both ADP and Pi dissociate from the active site and ATP rebinds to the active site. The forward and reverse rateconstants associated with this reaction are labeled gf and gr , respectively. Rates involving ff and fr do not involve the free energy of ATP hydrolysis, whereas those involving gf and gr do. Equilibrium constants, defined here in terms of rateconstant ratios, are related to free-energy differences.38, 39 For reactions in which ATP-associated energy is not involved,   ff (x) E0 − E1 (x) = exp , fr (x) kT where E0 and E1 (x) are the Helmholtz free energies of the head when unattached and attached at x, respectively, from actin. k denotes the Boltzmann constant and T the absolute temperature. For the reaction involving ATP, gf (x) gr (x) where  =

μT −μD , μT kT

= e− ,

and μD are chemical potentials of E −E (0)

ATP and ADP, respectively, and  = 0 kT 1 is the energy difference between attached and unattached states at x = 0. The free energy stored in the head, when attached at x, equals the free energy of the attached state at the equilibrium displacement and a shortening force F1 (x) integrated over x. Since F1 (x) scales linearly with a Hookean stiffness paramdE eter b associated with the myosin head,38 F1 (x) = dx1 = bx. Consequently,  x 1 F1 (t)dt = E1 (0) + bx 2 . E1 (x) = E1 (0) + 2 0

Under steady-state conditions, Eq. (1) becomes a function of x only, −V

dn = [ff (x) + gr (x)][1 − n(x)] dx − [fr (x) + gf (x)]n(x), x ≤ h,

(2)

where V = −dx/dt is the shortening velocity of a myosin head. Following Huxley,37 Eq. (2) was solved under two conditions, called Case (a) and Case (b). Case (a) occurs when 0 < x ≤ h, ff (x) = f1 xh , and gf (x) = g1 xh . Here, f1 and g1 are particular first-order rate-constants associated with the attached and detached rates, respectively. Case (b) occurs when −∞ < x < 0, f(x) = 0 (no attachment), and g(x) = g2 (constant detachment). Here, a myosin head is assumed not to attach at negative x. Huxley37 determined f1 = 43.3 s−1 , g1 = 10 x−1 , and g2 = 209 s−1 by fitting Hill’s T − V equation to frog muscle data.60 We fixed b = 3.0 pNnm−1 , which is approximately the average of the empirically determined values, 0.5– 7 pNnm−1 .43, 77 At 300 K, kT = 4.14 pNnm. n(x) should satisfy the condition n(x) = 0 for x ≥ h, and continuity between adjoining regions, such that n(0+ ) = n(0− ). Note that n(h) = 0, and this provides the initial data for Eq. (2). Within these constraints, the given conditions allowed us to obtain the solutions algebraically when V = 0, and in terms of integrals that can be approximated using numerical integration when V = 0. 1) For V = 0 and x < 0, pe−ax b , where a = 2 −ax 2kT 1+pe 2

n(x) =

and p = e− . (3)

2) For V = 0 and 0 ≤ x ≤ h, f1 + g1 pe−ax n(x) = . f1 + g1 + f1 eax 2 − + g1 pe−ax 2 2

(4)

3) For V = 0 and x ≤ h, we solved Eq. (2) under the initial condition n(h) = 0, obtaining  h 1 n(x) = es(x) e−s(z) [ff (z) + gr (z)]dz, (5) V x x where s(x) = V1 0 [ff (t) + fr (t) + gf (t) + gr (t)]dt. Recall that ⎧ ⎨ f1 xh + g1 p xh e−ax 2 , 0 ≤ x ≤ h ff (x) + gr (x) = ⎩ g pe−ax 2 , x h, n(x) = 0. Once n(x) is determined, we obtained the average force per myosin head by dividing the work of a myosin head by d,39 as 

  h 0 b Fm = n(x)xdx + n(x)xdx . (7) d d 0 2 In this case, muscle tension (Tm ) becomes MSm F m, (8) 2 where M denotes the number of myosin molecules per volume of one half-sarcomere and Sm denotes the length of a sarcomere. Tm =

Myosin heads generate this tension only when bound to actin filaments, which is no more than ca. 10%–20%,71 and no less than 1%78 of the total cycle time. When V = 0, we have the maximum force (F0 ) and tension (T0 ). E. Ring contraction tension

Before justifying a possible T − V relationship for yeast, we assume that a reasonable weighting factor compensating for the different activities of the actin and myosin components in muscle and yeast should be determined. Then, the suggested SSL structure of the ring [Fig. 1(c), right] is also expected to follow a relationship similar to Eq. (8), which describes muscle sarcomere contraction. Thus, we need to modify Eq. (8) to obtain an expression appropriate for ring tension (Tr ) by introducing a weighting factor [or reduction factor (RF)] and some different constriction dynamics between muscle and yeast [Eq. (9)]. This modified equation explains why ring constriction velocity is so much slower than that of muscle; it is evident that the number of myosin molecules in yeast would be much smaller than that in muscle (here, we presume that this may be a critical factor in their differences). However, to compare the contractile efficiencies that may originate from the different physiological properties of muscle and yeast, we assume that a sarcomere in muscle has the same number of myosin molecules as that of a hypothetical sarcomere in yeast. Thus, determining the RF may depend, in part at least, on the following four candidates. (1) Effect of α-actinin alignment rate: The concentration of α-actinin affects ring tension because it influences the degree to which actin filaments are aligned.20 Bendix et al.21 found that in an in vitro animal actomyosin network, contraction commenced when the α-actinin/actin molar ratio was > 0.05 and that the contraction velocity was maximized when this ratio was 0.14.

(2) Effect of Actin Filament Orientation: The SSL structure, with a mixture of viable and non-viable MCEs, would generate less tension than that of muscle, where all actin filaments are oriented viably. The SSL ring assembly mechanism suggests half of each orientation as a lower limit, whereas Carlsson’s contractility data69 suggest that the ring may have a higher proportion of viable sarcomeric structure. We need to consider this factor, which is unquantifiable in our current limited knowledge. (3) Effect of Imperfect Phasing: An imperfect phasing or register of actin:myosin pairs in adjacent MCEs may slow contraction of the actomyosin ring. Rings do not have periodical intervals with precisely arranged9 rows and columns. This lack of register would make it more difficult for a myosin filament to be effectively coordinated by surrounding actin filaments. (4) Effect of Septum Formation: Ring constriction may be coupled with septum formation in fungi because the cell wall and the membrane are bound.9 This may reduce Tr . We have ignored the effect of the myosin/actin molar ratio on ring tension in our model because ring tension depends only on the number of actin filaments, not on their length. The number of actin filaments is constant,13, 18, 30 even when each filament shortens. This approach resolves Schroeder’s dilemma, who postulated that the sliding filament mechanism only functions when the number and length of actin filaments are both invariant.28 We have ignored the effect of ring curvature on ring tension due to its negligible effect (Appendix B). When these reduction factors and the newly developed ring dynamics in yeast are combined, ring tension can be expressed as Tr (y) = N C(y)RF

yNh−MCE Sr F m, 2

(9)

where NC(y) denotes the normalized contractility,79 which was introduced to represent the phenomenon that myosin heads become inefficient when working against other attached heads (Sec. III E.), y denotes the number of myosin molecules per half-MCE (consisting of one actin filament and half of a myosin bipolar filament), Nh−MCE denotes the number of halfS MCEs per volume π r 2 2r of one half-SSL unit in wild type (WT) S. pombe (r denotes half of the ring thickness and Sr denotes the variable length of a SSL unit). In the equation for Tr (y), the fixed length of Sm in Eq. (8) was replaced with Sr . Also RF was adjusted so that the theoretically calculated decline [Eq. (14)] in ring radius during constriction matches what is observed experimentally. When the dimensionality of the torus ring is reduced to a 1D circle, the volume of one S S S half-SSL unit π r 2 2r reduces to 2r , such that yNh−MCE 2r becomes the number of myosin molecules. This explains why ring tension is insensitive to the length of the decreasing ring SSL unit (or ring radius), but is proportional to the number of myosin molecules in the ring. For simplicity, the number of myosin molecules in our model was assumed constant during ring constriction, which results in constant ring tension. However, in WT cells, the number of myosin molecules gradually

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J. Chem. Phys. 141, 125101 (2014)

decreases, nonetheless, the concentration of myosin increases 2.5–3 fold during a 20 min ring constriction8 [Fig. 4(d) and Appendix C].

The ring constriction velocity equation is obtained by differentiating Eq. (14), affording

F. Ring constriction tension and velocity

Substituting the time of ring constriction, 1380 s,4, 85 into Eq. (14) and setting the final ring radius (before the ring disappears) at R0 /124, 85 affords   KR 0 12 −Tt ln KR −Tt 0 , (16) ξ= 1380K

The two tensions involved in ring contraction include the constriction tension Tr (y) and the opposing elastic resistance tension (ERT).35, 80 Constriction tension mainly arises from the power stroke (and possibly from actin filament dynamic exchange), whereas ERT arises from the membrane resisting ring constriction.19 We calculated ERT using a modified Yoneda equation80 [Eq. (10)]. The Yoneda equation has been used to calculate ERT in animal cells by predicting the minimal constriction force required to pull the membrane inward. ERT was estimated by relating the constriction force to the ring radius and the degree of ingression of the cell. We modified the original Yoneda equation so that it would be applicable to S. pombe. This organism maintains a cylindrical shape during cytokinesis, whereas the shape of animal cells changes from spherical to dumbbell-shaped as the ring constricts. This straight vertical division with no degree of ingression in S. pombe affords the relationship Ter = −KR(t),

(11)

where 2π Tr denotes the total circumferential ring tension Tt normal to the ring surface35 and λ denotes the tension generated by the turnover rate (or accelerated depolymerization). Ring constriction velocity Vr equals Tr divided by an effective friction coefficient ξ −1 (unit: pNs nm−1 ),35, 46, 74 Vr = −ξ Tr .

(12)

Substituting (11) into (12) and setting λ ∼ = 0 (Sec. II C.) affords the velocity equation dR(t) = −ξ [−KR(t) + Tt ]. dt Solving this equation yields   Tt T E ξ Kt + t . R(t) = R0 − K K Vr (t) =

(15)

which relates ξ to K. III. RESULTS

Main results of this study include: (1) determination of ring tension using a self-consistent physicochemical parameters set of , , V , and RF ; (2) demonstration that ring constriction velocity is an exponential function of time despite an appearance that seems to be almost linear; (3) demonstration of how myosin concentration and elastic resistance tension are balanced; and (4) demonstration of how doubling the myosin concentration reduces ring constriction velocity.

(10)

where R and K denote the ring radius and the elastic modulus, respectively. The elastic modulus reflects membrane stiffness. When determining the ERT-associated reduction in the rate of ring constriction, we only considered the ERT generated by the membrane K and ignored the huge elastic modulus of the cell wall (ca. 20 nN nm−1 ).81 The cell wall is not thought to be bent inwards by the ring, but will instead remodel and ingress as the ring itself drives constriction. Nevertheless, we have included the effect of septum formation as a component of RF, in that the coupling between septum formation and ring constriction may reduce the rate of cytokinesis.9, 30 Typical K values in different organisms range from about 0.01 to 10 pN nm−1 .35, 82–84 Since the ring constriction tension squeezes S. pombe uniformly around the circumference in the radial direction and opposes ERT, the net ring tension (Tr ) equals the sum of three different constriction tensions Tr = −KR(t) + 2π Tr + λ,

Vr (t) = ξ (KR0 − Tt )eξ Kt .

(13)

(14)

A. Numerical simulations

1. Parameters set

Numerical simulations were performed using MAPLE (version 16, http://www.maplesoft.com) and MATLAB (version 7.12.0.635, http://www.mathworks.com). The values for attachment and detachment rates and the Hookean stiffness parameter of the myosin head were adopted from muscle contraction data, whereas the number of cytokinesis proteins assumed was based on experimental results.8 Equations (3)–(8) show how muscle tension depends on n(x), which is restricted by the interrelationships between  (the hydrolysis energy),  (the free energy difference between the attached and unattached states), and V (the shortening velocity of actin:myosin interactions). We found a selfconsistent set of parameters  = ln (109.5 ),  = ln (108.4 ), and V = 69 nm/s that are physiologically possible, and in our view, most realistic. These parameters were used in all subsequent numerical simulations. Whether constriction times are proportional to or independent of ring size is uncertain. Experimental results suggest that they may be proportional,4, 79, 85 whereas the constriction models of Carvalho et al.30 (Carvalho et al. proposed that the length of actin filaments in C. elegans which may determine constriction times is independent on ring size) and Pinto et al.72 (in budding yeast) suggest that they be independent. In our model, we assumed that constriction times (in fission yeast) are proportional to ring radius because large ring radii would have longer actin filaments, which need more time to constrict the ring. Under this assumption, and for the purpose of comparison, experimental ring radii and constriction times used in this study were normalized such that a ring with a 1.64 μm radius constricted in 23 min in our simulations. Note that our constriction model can be modified when more exact

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J. Chem. Phys. 141, 125101 (2014)

TABLE I. Calculated muscle and ring tensions when  = ln (109.5 ). Values were obtained using Eqs. (1)–(9) and different values of Z in  = ln(10Z ) (F0 = force when V = 0, T0 = tension when V = 0, Vmax = maximum velocity, Fm = force (in muscle-like structure), Tm = tension (in muscle-like structure), Tr = ring tension, Tt = total circumferential ring tension). Z

6 7 8.4 10

F0 (pN)

T0 (pN)

Vmax (nm/s)

Fm (pN)

Tm (pN)

Tr (pN)

Tt (pN)

1.3 1.5 1.6 1.5

240 270 300 270

750 760 690 300

0.86 1.0 1.1 1.2

160 180 190 220

5.9 6.7 7.1 8.1

37 42 45 51

information on the length of actin filaments in many different species becomes available. 2. Determination of RF

Equation (15) was used to obtain an initial constriction velocity Vr (0) = ξ (KR0 − Tt ). To constrict the ring, Tt should be >KR0 which requires ca. 16.4–16 400 pN, when K ranges from 0.01 to 10 pN nm−1 , respectively. Ring constriction behavior would likely be similar even if the actual K were outside of this range. When K = 0.01 pN nm−1 , Tt ∼ = 45 pN (Tr ∼ = 7.1 pN) was the best-fit value obtained by substituting random Tt values into Eq. (14) to match the normalized experimental data shown in Fig. 4(c), circles.4 The least squares objective function used was similar to Eq. (18). The Tm associated with a ring consisting of perfect muscle-like sarcomeres was ca. 190 pN (Table I). For Tt = 45 pN, this implies that RF ∼ = 3.7 × 10−2 (the sum of effects of four reduction factors and perhaps other unidentified factors), which was assumed for all downstream simulations. We assumed that ξ would be similar in both the muscle and the ring [Eq. (12)]. This Tt is equivalent to the contraction force that can be generated by less than ca. 1% of total myosin molecules in S. pombe, calling into question whether it is sufficient to constrict the ring. However, this estimate is consistent with the fact that myosin with only 2% of WT activity suffices for S. pombe cytokinesis.58 In addition, another theoretical model predicted that a tension of 8 pN is sufficient to cause Escherichia coli to divide.86 Very recently, Stachowiak et al.87 reported that the experimental ring tension of S. pombe is ∼390 pN. Thus, we regard these estimates of a smaller K and Tt to be comparatively more reasonable, which implies that K  10 pN nm−1 . We also evaluated the sensitivity of this parameter by determining the effect of doubling it (K = 0.02 pN nm−1 ). B. Determining , , and V by constraining n(x)

Calculating ring tension required that we first obtain the attachment probability n(x) of a myosin head, which is a function of , , and V . Experimentally determined in vivo values of  range from ln (108 ) (μT − μD = 11 kcal mol−1 at 300 K) to ln (1011 ) (μT − μD = 15 kcal mol−1 at 300 K),39 whereas the value of  is unknown. We set the value of  at ln (109.5 )

which is the mean value between ln (108 ) and ln (1011 ) . To find a physiologically relevant value of , we adopted the empirical T − V equation of Hill (1938) in which empirical muscle tension Tm (i) [where i varies from 1 to V max (maximim velocity) in increments of 1] satisfies the following relationship: Tm (i) =

(Vmax − Vi )Tmax . Vmax + 4Vi

(17)

Once  was selected, we calculated both Tm (simulation) and Tm (Hill) using the same Vmax and Tmax (maximum tension) generated by . Then, we have the following least-squares objective function ME (mean error): Vmax [Tm (i) − Tm (i) ]2 . ME = Vmax i=1

(18)

The value of  = ln (108.4 ) afforded the minimum ME. The pair [ = ln (109.5 ) and  = ln (108.4 )] along with Eqs. (3)– (8), (17), and (18) [Fig. 3(a), red line] reproduced the empirical T − V equation (circles) of Hill (1938). This [, ] pair was used for all subsequent simulations. The hyperbolic T − V curve shows that lower velocities for myosin heads, where more myosin heads are attached to actin filaments, generate higher muscle contraction tension. Using the selected [, ] pair and Eq. (7), we found that Vmax = 690 nm/s when F = 0. The optimal shortening velocity of the head, V /Vmax , was set at 0.1088 such that V = 69 nm/s. This V /Vmax was chosen to match the minimum force of 1–2 pN37, 89 experimentally determined in actin:myosin interactions. In muscle contractions, maximum power [i.e., energy/s = (tension × distance)/s = tension × velocity] occurs at the optimal shortening velocity. Simulations using the selected [, ] pair revealed that the shortening velocity is inversely proportional to n(x) [Figs. 3(b) and 3(c)]. When V = 0, a myosin head has an equal probability of attaching to actin over the range of displacements between 0 and 7, resulting in a near rectangular plot [Fig. 3(c), plot i]. When V = 0, the head moves down into the negative-force zone as the shortening velocity increases, gradually reducing the area associated with the positive-force zone [Fig. 3(b)]. If g2 is reduced, the contraction force becomes negative because the head is pulled too much into the negative zone. When the areas of negative- and positive-force zones are equal, the head will detach from an actin filament by reaching Vmax [Fig. 3(c), plot iv]. This shows that Vmax is related to the rate of ADP release, which leads to dissociation of myosin from actin.90 When the muscle-based T − V relationship was analogously applied to the ring, the higher n(x) with lower shortening velocity generated higher ring contraction tension. Higher ring contraction tension will increase constriction velocity, as described by Eq. (13). C. Simulations of ring constriction

The selected set of , , V , and RF of ca. 3.7 × 10−2 yielded an actin:myosin interaction force of 1.1 pN and a ring tension of 45 pN (Table I). Calculated ring tensions obtained

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Y.-W. Jung and M. Mascagni

J. Chem. Phys. 141, 125101 (2014)

1

1

0.9

0.9 1

0.8

0.7

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n(x)

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125101-10

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i ii

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iii

0 0.2

0

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V (nm/sec)

T/T0

0.1

−5

200 0

iv

5

400

0.1 0

0.2

600

−10 0

0

−15

x (nm)

−15

−10

−5

(b)

(a)

0

5

x (nm)

(c)

FIG. 3. T − V and n(x) curves. (a), T − V curves generated using Eqs. (3)–(8). (b), graph of n(x) displacement position x generated using Eqs. (3)–(6). V ranged from 0 to Vmax = 690 nm/s. (c), same as (b) but plotted at V = (i) 0, (ii) 0.1Vmax , (iii) 0.5Vmax , and (iv) Vmax ( nm/s).

using several different Z values (6, 7, 8.4, and 10) were not significantly different from each other (Table I). We investigated how Tr , ERT, and Vr are interrelated. During a ring constriction simulation using K = 0.01 pN nm−1 and Tt = 45 pN, Tr [Fig. 4(a), plot ii] increased in accordance with a gradual decrease in the ERT of S. pombe [Fig. 4(a), plot

iii]. This means that the ring can be squeezed with increasing ease as its radius declines, as described by the modified Yoneda Eq. (10). To evaluate the sensitivity of K in the process of ring constriction, Vr and Rr (ring radius) were compared using two different K values (0.01 pN mn−1 and0.02 pN nm−1 ). Vr var-

50

−0.5

i

40

ii

30

i

V (nm/sec)

20

10

ii

r

T (pN)

−1

−1.5 0

−10

iii −20

0

200

400

600

800

1000

−2

1200

0

200

400

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600

(a)

1600

450

Protein concetrations ( µM)

500

1400

r

R (nm)

1200

i

ii

800 600 400

300 250 200 150 100

0

0

400

600

800

1000

1200

iii

350

50 200

1200

400

200

0

1000

(b)

1800

1000

800

Time (sec)

i

ii iv

0

200

400

600

800

Time (sec)

Time (sec)

(c)

(d)

1000

1200

FIG. 4. Calculated ring tensions, constriction velocity, and radius. (a), i is the constriction ring tension, iii is the ERT, and ii is the net ring tension Rr equal to the sum of i and iii. (b), simulated constriction velocities Vr using K = 0.01 (i) and 0.02 (ii) pN nm−1 . (c), ring radii Rr , calculated using Eqs. (3)–(16), and K values as in (b). The data (circles) were reproduced from Pelham and Chang4 and were normalized for comparison to our simulations. (d), simulated protein concentrations, including (i) myosin when K = 0.01 pN nm−1 , (ii) myosin when K = 0.02 pN nm−1 , (iii) actin, and (iv) α-actinin.

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J. Chem. Phys. 141, 125101 (2014)

ied from −0.5 to −2 nm/s during ring constriction [Fig. 4(b)]. The resulting pattern was similar to that of Dictyostelium, which varied exponentially with time.84 Use of K = 0.01 pN nm−1 and Tt = 45 pN reproduced the normalized ring contraction data obtained from fluorescence microscopy [Fig. 4(c)].4 The decline in ring radius roughly appears linear [Fig. 4(c), circles],4, 8, 58, 59, 85 but in fact is exponential [Fig. 4(c), plot i] as our model predicts. The slow initial decline is probably due to the ERT. Ring radius, as calculated [using Eqs. (9), (14), and (16)] from the constriction radius graphs, approached R0 /12 in 1380 s [Fig. 4(c)]. When K was increased 2-fold, the simulation did not overlay the data [Fig. 4(c), plot ii], indicating the sensitivity of the elastic modulus in S. pombe cytokinesis. In contrast, calculated velocities were rather insensitive to the values of f1 , g1 , and g2 (data not shown), implying that the exact values assumed for these parameters were not critical. While the simulated concentrations of actin and αactinin were constant, that of myosin was gradually increased, although the number of myosin molecules roughly decreased [Fig. 4(d), plots i and ii] (Appendix C), as observed experimentally.8 Note that the theoretical estimate of myosin concentration beyond 20 min after ring constriction does not match the experiment due to unquantifiable ring abscission. With Tr constant, the effective frictional coefficient (ξ −1 ) decreased as K increased. The value of ξ is unknown so it was estimated from the assumed K value [Eq. (16)].

FIG. 5. Real and simulated images of ring constriction. (a) time-lapsed images of ring constriction in WT S. pombe [Reprinted with permission from R. J. Pelham, Jr. and F. Chang, Nature (London) 419, 82–86 (2002). Copyright 2002 by Macmillan Publishers Ltd.]. (b) and (c), simulations generated using K = 0.01 pN nm−1 and 0.02 pNnm−1 , respectively. Simulations assumed the same parameters used to solve Eqs. (9), (14), and (16).

[Fig. 5(c)] did not match the real time-lapsed images, confirming the high sensitivity of the elastic modulus in cell cytokinesis. E. Effect of excessive myosin concentration on ring constriction velocity

Stark et al. recently found that doubling the myosin concentration in fission yeast, relative to WT levels, reduced ring contraction velocity to ca. 70% of the WT value.79 Doubling the myosin concentration may increase the dynamic exchange rate of myosin filaments, which has been suggested to cause nonproductive filament contractility.4, 29, 79 The phenomenon may be explained that excessive myosin heads would generate forces opposing contraction,91 by reducing the space available for sliding (steric crowding). In this situation, we hypothesize that excessive myosin would lead to less actin filament sliding79 due to the holding forces of other attached myosin heads waiting for power strokes. This possibility requires a fundamental assumption: Myosin heads act independently in attaching to and detaching from actin, and in initiating the power stroke. We demonstrated this hypothesis using the binomial theorem. Fig. 6 shows how we reproduced the

D. Simulation of time-lapsed images of ring constriction

Time-lapsed microscopy of the actomyosin ring in fission yeast [Fig. 5(a)] illustrates the kinetics of ring constriction and the invariant ring thickness.4 In these images, the ring, starting with a radius of 1.85 μm , constricted in 1500 s. To illustrate the decreasing rate of ring radius in our ring model in 3D, we simulated a torus with the same initial radius (1.85 μm) and invariant ring thickness (0.173 μm). Using Eqs. (9), (14), (16), and K = 0.01 pN nm−1 , we reproduced the approximate experimental rate of ring constriction [Fig. 5, (a) vs. (b)]. The simulated rate of contraction using K = 0.02 pN nm−1 1 1

0.8

r

0.7 0.6 0.5 0.4

0.6

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1400

0.2

1200

Rr (nm)

Normalized V (nm/sec)

0.9

NC

1800

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60

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(a)

80

100

ii

200 1000

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0

20

(b)

40

60

Number of myosin (y)

80

100

0

0

500

1000

1500

2000

2500

Time (sec)

(c)

FIG. 6. Inverse effect of myosin on constriction velocity. (a) normalized contractility vs. number of myosin molecules per actin filament. Red rectangle and blue circle indicate WT and mutant cells, respectively. (b) normalized ring constriction velocity Vr . (c) ring constriction radius Rr . WT and mutant data were adapted with permission from Stark et al., Mol. Biol. Cell 21, 989 (2010). Copyright 2010 by The American Society for Cell Biology, and normalized for comparison. Simulations were generated using K = 0.01 pN nm−1 and Eqs. (10)–(18).

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experiment based on the idea suggested in Appendix A. Setting U = 0.971 in NC(y) (Appendix A) and assuming that ca. 10 myosin molecules per actin filament afford optimal contractility (whendC/dy = 0) gives A = 0.09 and P = 0.01 (i.e., the head is in the power-stroke state 1% of the time). In these conditions, both the WT contractility [Fig. 6(a)] and the WT constriction velocity [Fig. 6(b)] decreased to ∼70% when the myosin concentration increased two-fold. Simulations i and ii in Fig. 6(c) reproduced the experimentally observed decline (open and filled circles).79 Interestingly, doubling myosin increased the ring assembly rate twice.79 This implies that the nodes used in assembly may increase the number of myosin filaments, while maintaining the number of myosin heads per filament, although the detailed mechanism is unknown.

J. Chem. Phys. 141, 125101 (2014)

(15), and Figs. 5(b) and 5(c)]. Our statistical analysis offers a new explanation as to why doubling the myosin concentration might reduce ring constriction velocity. The quantitative model developed here for the ring in fission yeast cells should apply to the mechanism of actomyosin ring constriction in animal cells. A quantitative understanding of this may provide insights into human-cell cytokinesis and the other aspects of the cell cycle process. This could have downstream implications for understanding the mechanism of cancer and other cell-cycle-related diseases. ACKNOWLEDGMENTS

The authors are grateful to Dimitrios Vavylonis, Jian-qiu Wu, and Fred Chang for their helpful comments.

IV. CONCLUSIONS AND DISCUSSION

For three decades, the actomyosin ring has been presumed to constrict by a sliding filament mechanism,57 but the implications of this have not been explored quantitatively. Our efforts to do this were prompted by Wu and Pollard’s recent experimental determination of the stoichiometric concentrations of the components of the ring of fission yeast8 and the mechanism for ring assembly proposed by Vavylonis et al.5 These elegant studies provided a foundation to consider possible molecular-level ring structures (and assembly) that could constrict by the sliding filament mechanism. We also adopted the Huxley-type Hill sliding-filament formalism into our ring model, which allowed us to estimate ring constriction tension (when K is given) and reproduce ring constriction velocity in S. pombe. Although many of the required parameters for simulating ring constriction have not been determined experimentally, analogous parameters associated with muscle contraction are available and were employed. Other parameters were constrained, ultimately affording a physiologically possible, self-consistent set of values, including n(x), ff , fr , gf , gr , , , V , the ring constriction tension, and the resistance tension that simulated observed ring constriction velocities (Figs. 3–6). We proposed several reasons why the constriction velocity of the ring in fission yeast is a thousand-fold slower than that of muscle sarcomeres. These included the effects of the number of myosin molecules involved, α-actinin alignment, filament orientation, the register of laterally-associated contractile elements, and the cell wall septum. Using the framework established here, more constrained and increasingly unique simulations are expected to reduce the uncertainties (especially, a relationship between septum formation and ring constriction) as more experimentally determined parameters become available. In addition, it may be necessary to investigate how the ATP hydrolysis activity of ring and muscle differentiates the rate of contraction velocity. ERT was also a significant factor contributing to slow ring constriction and sensitively reacting to cell cytokinesis (Figs. 4 and 5). Our analysis demonstrates that the constriction velocity of S. pombe is an exponential function of time, contrary to the previous expectation that it declines linearly [Eqs. (13) and

APPENDIX A: MATHEMATICAL FORMULATION OF RING CONTRACTILITY

Normalized contractility [NC(y)] was introduced to represent the phenomenon that myosin motors become inefficient when working against other attached motors. This indicates that ring activity has evolved over billions of years to retain the optimal number of myosin motors that maximizes its energy efficiency. Here we introduce a possible mechanism in a statistical point-of-view. Actin:myosin head interactions can be divided into three sequential states, including the power stroke (with probability P), an unattached state (U), and a static attached state (A). So the state P in one myosin head sliding along one actin filament is assumed to generate a contractile force only when the other myosin heads simultaneously sliding along with the same actin filament are in the states P and (or) U. This assumption implies that excessive heads interacting with the same filament reduces contractility (C) because the probability becomes very high that at least one head will be in the A state when another is in the P state. The following binomial theorem shows that C depends on the combination probability of P and U: C=

y−1   y P y−k U k = (P + U )y − U y , k k=0

where y denotes the number of myosin molecules per actin filament. Then, the normalized contractility79 [NC(y)] is expressed as N C(y) =

[(P + U )y − U y ]/y , [(P + U )yW T − U yW T ]/yW T

where yW T denotes the number of myosin molecules per actin filament in WT S. pombe. APPENDIX B: EFFECT OF RING CURVATURE ON RING TENSION

Ring curvature introduces a deviation from parallel actin filaments, from one ring SSL unit to the next (Fig. 7). Angle ( ABC) is 11.25◦ , which will be the maximum angle between the tangent line and the ring circle. The maximum contraction

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J. Chem. Phys. 141, 125101 (2014) 12 T. D. Pollard, L. Blanchoi, and R. D. Mullins, Annu. Rev. Biophys. Biomol.

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FIG. 7. Ring curvature. In a half-SSL unit, there are the corresponding arc and cord connecting A and B, and C is the left point of a tangent line crossing point B.

force is F = F/cos θ , where F is the contraction force calculated in our ring model. F is ≈ F because cos 11.25◦ ∼ = 0.98. The same relationship between F and F holds as the ring radius decreases. APPENDIX C: CHANGES IN PROTEIN CONCENTRATIONS DURING RING CONSTRICTION

Initially, the torus ring has 66 700 actin monomers per ring volume 2π 2 R0 r2 (= π r2 × 2π R0 ). Because the ring constricts and its volume declines, we assumed that nA (t) =

66, 700R(t) , R0

where nA (t) is the number of actin monomers in the ring at time t. As long as the ring thickness remains unchanged, the concentration of actin [A] (unit μM) will also be invariant [A](t) ∼ = ∼ =

nA (t) μm3 × 2 2 2π R(t)r 602 111μm3 ∼ = 460. 2π 2 R0 r 2 (∼ = 0.24μm3 )

A similar situation holds for α-actinin, namely, that the number of molecules in the ring will decline as the ring constricts, but its concentration (4 μM) remains invariant. However, the concentration of myosin [M](t) (unit: μM) increases as the ring constricts, which is expressed as [M](t) ∼ = nM (t) =

μm3 nM (t) , where × 2π 2 R(t)r 2 602 2900[0.6 + 0.4R(t)] . R0

Here, nM (t) is the number of myosin monomers in the ring at time t. Note that the ring tension is proportional to the number of myosin molecules, not to the myosin concentration. 1 D.

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