A dynamical systems approach to actin-based motility in Listeria monocytogenes
aa r X i v : . [ q - b i o . S C ] N ov A dynamical systems approach toactin-based motility in
Listeria monocytogenes
Scott Hotton
Department of Organismic and Evolutionary Biology, Harvard University, Cambridge MA 02138
A simple kinematic model for the trajectories of
Listeria monocytogenes is generalized to a dy-namical system rich enough to exhibit the resonant Hopf bifurcation structure of excitable mediaand simple enough to be studied geometrically. It is shown how
L. monocytogenes trajectories andmeandering spiral waves are organized by the same type of attracting set.
Introduction.
Listeria monocytogenes is a widely dis-tributed pathogenic bacteria which occasionally causesserious illness in humans.
L. monocytogenes evades thehost’s immune system by living inside its cells. Proteinslocated on the surface of the rod shaped bacteria catalyzethe polymerization of the infected cells’ actin moleculesand this activity propels the bacteria through the cy-toplasm [1]. The underlying mechanism of actin-basedmotility is a subject of great interest both because
L.monocytogenes is a deadly pathogen and because actinfilament assembly plays a role in many forms of cellmovement [2]. A useful feature of actin-based motility in
L. monocytogenes is the “comet tail” of actin filamentswhich are left behind as a cell is transported [3]. The“comet tails” provide a record of bacterial trajectoriesin the cytoplasm. These trajectories, which vary con-siderably between individuals, can be complicated andorderly at the same time. In the present work it is shownhow these trajectories can be explained with a dynami-cal system that has a well known type of attracting set.The presence of this attracting set can account for the re-newal of actin-based motility after the bacterium dividesand the persistence of motility as the bacterium invadesneighboring host cells.
L. monocytogenes trajectories are the result of a com-plicated interaction of proteins in a host cell’s cytoplasmor in a cytoplasmic extract [4]. There has been extensiveresearch into the molecular mechanism underlying actin-based motility but they do not account for the compli-cated forms of
L. monocytogenes’ actin comet tails [5–12]. In [13] Shenoy et al. present a simple and remark-ably effective kinematic model for the trajectories of
L.monocytogenes in a thin layer of cytoplasmic extract.In the Shenoy et al. model the effect of actin poly-merization is assumed to produce a net force on the cellbody which points slightly off center and which causesthe bacteria to spin about its long axis as it travels intwo dimensions. The bacterium’s velocity is given by avector whose direction varies sinusoidally with time andwhose magnitude is fixed. Choosing units of measure sothe speed is 1, letting s stand for arc length, and θ standfor the velocity’s direction the Shenoy et al. model, ina non-dimensionalized form, is dθ/ds = Ω cos( s ) whereΩ ≥ Ωκ FIG. 1: Six curves in the ( x, y )-plane determined by dθ/ds =˘ κ + Ω cos( s ). The inset for each ( x, y ) curve points to its cor-responding parameter values (Ω , ˘ κ ). Each ( x, y ) curve startsat the point (0 ,
0) (marked by an open circle) in the direction θ = 0. In two cases with ˘ κ = 0 the curves exhibit linear drift(which follows the dotted lines). The paths are qualitativelythe same for ˘ κ = 1 /
20 but show an overall tendency to veerfrom a straight course (as indicated by the dotted curves).The region of parameter space shown here only contains aportion of the parameter values that can replicate
L. mono-cytogenes trajectories. motion.Since dθ/ds equals curvature the non-dimensionalizedform of the Shenoy et al. model gives a one parame-ter family of intrinsic equations for planar curves ( i.e. atwo dimensional analog for the Frenet-Serret equations)which exhibit qualitative changes as the parameter Ω isvaried. Shenoy et al. show that for small Ω the curveis sinusoidal, for Ω ≈ . et al. show that L. monocytogenes display most if not allof these types.Although this kinematic model can reproduce manyof the complicated
L. monocytogenes trajectories it doesnot explain how the forces on the cell body arise or ad-dress the stability of the motion. The question of howactin-based motility arises is important because the bac-teria have to divide in order to proliferate inside the hostorganism and after cell division the protein catalyzingactin polymerization is redistributed on the bacterial sur-face [14, 15]. The issue of stability is important becauseone of the functions served by actin-based motility is toenable
L. monocytogenes to create “pseudopodal projec-tions” from one host cell into another and thus allow thebacteria to avoid the host’s immune system [1]. If thedynamics underlying actin-based motility was not stablethen obstacles in the bacteria’s path could disrupt thebacteria’s entry into the neighboring host cell.In the next section the model of Shenoy et al. is ex-tended to allow the angular displacement of
L. mono-cytogenes trajectories to accumulate. The third sectionexplains how this extension leads to a dynamical systemwith the same type of attracting set as seen with mean-dering spiral waves. The fourth section shows how theexistence of the attracting set accounts for actin-basedmotility developing and persisting in
L. monocytogenes .
2. Generalizing the kinematic model.
It is wellknown in ballistics that it is difficult for a projectile totravel in a perfectly straight direction. For a self pro-pelled bacterium in a viscous medium even a small asym-metry in the cell body can cause it to eventually deviatefrom a straight course [16]. On the other hand in theShenoy et al. model the angular displacement remainswithin fixed bounds along the entire length of the curveand the overall trajectory conforms to a straight line de-spite the small scale oscillations in its direction.To improve the accuracy of the kinematic model it isworth considering previous studies on actin-based motil-ity. Rutenberg and Grant [17] related the overall cur-vature of the trajectories to the number of randomly lo-cated actin filaments propelling the cell. They treatedthe torque produced by the filaments as a constant forrelatively long periods of time which led to trajectorieswith constant curvature, dθ/ds = ˘ κ . For trajectorieswith ˘ κ = 0 starting in the direction θ = 0 the angulardisplacement grows in proportion to arc length.Evidence for the secular dependence of angular dis-placement on arc length in the Rutenberg and Grantmodel was subsequently presented in the extensive ex-perimental study on actin-based motility in L. mono-cytogenes [18]. The study found bacteria trajectoriesthat were nearly circular and whose angular displace-ments and path lengths were nearly proportional to theelapsed time. Consequently the angular displacementswere nearly proportional to arc length.Here we combine the approaches of Shenoy et al. andRutenberg and Grant into a single model and present asummary of the types of paths this model displays. TheShenoy-Rutenberg model is, in non-dimensional form, dθ/ds = ˘ κ + Ω cos( s ) where Ω, ˘ κ are constants. Forsmall ˘ κ the paths are qualitatively the same as for ˘ κ = 0but they veer from a straight course (fig. 1). Shenoy et al. found it useful to modify their model in a few casesby adding a low frequency term but this still did not al-low the angular displacement to accumulate. Adding aconstant term as we do here does allow the angular dis-placement to accumulate and it helps shed light on theeffectiveness of the Shenoy et al. model.The Shenoy-Rutenberg model is comparable in form toa model by Friedrich and J¨ulicher for the chemotaxis ofsperm cells [19]. Both models determine the curvature ofthe path followed by the cells using a constant curvatureterm and a second term but they differ in the form of thesecond term. For the Friedrich and J¨ulicher model thesecond term is a function of the chemoattractant concen-tration and the internal signaling network. The pathsproduced by the Friedrich and J¨ulicher model depend onthe form of the concentration field.It is not very difficult to determine the paths producedby the Shenoy-Rutenberg model. Let ( x ( s ) , y ( s )) denotethe arc length parameterization of a path. The additionof ˘ κ leaves the model in the form of an intrinsic equationfor planar curves so, for fixed values of the parameters,the solutions are congruent and we can focus on the initialcondition ( x, y, θ ) = (0 , , (cid:18) x ( s ) y ( s ) (cid:19) = Z s (cid:18) cos(˘ κσ + Ω sin( σ ))sin(˘ κσ + Ω sin( σ )) (cid:19) dσ (1)Changing the sign of either Ω or ˘ κ yields congruent ( x, y )curves so we can assume Ω , ˘ κ ≥
0. While the integralcannot be evaluated in terms of elementary functions thecurves are symmetrical and made up of congruent copiesof an arc of length π . Since the ( x, y ) curve is invariantunder reflection about the y -axis we can reflect the arcfor 0 ≤ s ≤ π to obtain the arc for − π ≤ s ≤ π .For non-integral ˘ κ let r = cot( π ˘ κ ) x ( π ) + y ( π ). It canbe shown that (cid:18) x ( s + 2 π ) y ( s + 2 π ) − r (cid:19) = (cid:18) cos(2 π ˘ κ ) − sin(2 π ˘ κ )sin(2 π ˘ κ ) cos(2 π ˘ κ ) (cid:19) (cid:18) x ( s ) y ( s ) − r (cid:19) (2)From this it follows that the ( x, y ) curve can be obtainedby iteratively rotating the arc for − π ≤ s ≤ π aboutthe point (0 , r ) which is the center of symmetry for thefigure.For non-integral rational ˘ κ = p/q ( p, q coprime) andΩ = 0 the ( x, y ) curve is closed with q -fold rotationalsymmetry. It is the union of 2 q congruent arcs of length π . For irrational ˘ κ and Ω = 0 the ( x, y ) curve is quasiperi-odic in the plane. It is the union of an infinite numbercongruent arcs with length π .For integer values of ˘ κ we can think of r as having goneto infinity. It can be shown that y ( s + 2 π ) = y ( s ). Toexpress the value of x at multiples of π we can use theintegral representation for Bessel functions J ˘ κ ( − Ω) = 1 π Z π cos(˘ κσ + Ω sin( σ )) dσ (3)(this integral representation does not apply to non-integer values of ˘ κ ). From this it follows that the ( x, y )curve can be obtained by iteratively translating the arcfor − π ≤ s ≤ π horizontally by the distance 2 πJ ˘ κ ( − Ω).When − Ω is a zero of J ˘ κ the ( x, y ) curve is closed withlength 2 π . Otherwise the ( x, y ) curve is the union of aninfinite sequence of congruent arcs with length π . Thisgeneralizes a similar result from [13] for the ˘ κ = 0 case.The points of maximal curvature on an ( x, y ) curveoccur where s is an even multiple of π , the points ofminimal curvature occur where s is an odd multiple of π , and the curvature varies monotonically in between. Ina neighborhood of (0 ,
0) an arc of the ( x, y ) curve liesabove the horizontal tangent at (0 ,
0) and for ˘ κ >
Ω thecurvature is positive everywhere.For 1 = ˘ κ >
Ω the ( x, y ) curve has the form of atrochoid with its “petals” lying in a row (fig. 2). Forsmall Ω and 0 < ˘ κ < x, y ) curve has the form of ahypotrochoid with its “petals” on the outside. For smallΩ and 1 < ˘ κ < x, y ) curve has the form of anepitrochoid with its “petals” on the inside.Flower like curves such as these are traced out by thetips of spiral waves propagating through excitable me-dia. Spiral waves occur in diverse systems with very dif-ferent underlying mechanisms. This includes aggregatingmyxobacteria which form macroscopic waves as cells glideacross a two dimensional surface [22]. A spiral wave oftenpropagates as though it were a rigid body rotating abouta quiescent core. Away from the core the shape of thewave front converges to an Archimedean spiral [23, 24].However under appropriate circumstances the inner tipundergoes a secondary oscillation as the wave rotates andthereby traces a hypo/epi/trochoid like curve. Spiral tipmeander has been observed in many systems such as theBZ chemical reaction [25], heart tissue [26], and aggre-gating cells of Dictyostelium discoideum (cellular slimemolds) [27].
3. Resonant Hopf bifurcations.
An importantstep toward understanding why spiral tip meander oc-curs in systems with such different underlying mech-anisms was made by Barkley [28–30] who recognized,through numerical and mathematical analysis, the im-portant role played by the group of orientation preserv-ing congruences of the Euclidean plane, SE (2), and thatthis role can be exemplified by reducing the dynamicsto five dimensions. The mathematics of Barkley’s break-through has been further elaborated and generalized [31–35]. Barkley’s approach can be nicely illustrated with theShenoy-Rutenberg model since it already has the formof an intrinsic equation for planar curves. To do this wecouple the Shenoy-Rutenberg model to a two dimensionalsystem from [36]. We write the Cartesian coordinates forthis subsystem as ( X, Y ). The full differential equation κ Ω
FIG. 2: A version of a Zykov-Winfree flower garden [37, 38]whose isogonal contours have been combed straight. The isog-onal contours of dθ/ds = ˘ κ + Ω cos( s ) for ˘ κ = 2 / , / , , / x, y ) curve in the insets starts at the point(0 ,
0) (marked by an open circle) in the direction θ = 0. For˘ κ = 1 the ( x, y ) curves exhibit linear drift. For ˘ κ below 1 the( x, y ) curves have hypotrochoid like shapes and for ˘ κ above1 the ( x, y ) curves have epitrochoid like shapes. So long as˘ κ > Ω (above the diagonal line) the ( x, y ) curves do not haveinflection points. is x ′ = cos( θ ) y ′ = sin( θ ) θ ′ = ˘ κ + X (4) X ′ = − Y + ( µ − X − Y ) XY ′ = X + ( µ − X − Y ) Y where the parameter µ corresponds to Barkley’s normal-ized bifurcation parameter [29]. For µ < X, Y ) subsystem is an attracting fixed point. At µ = 0 the Hopf bifurcation occurs and for µ > √ µ .To concisely express the initial conditions we set Ω = 0for µ < √ µ for µ ≥
0. For the initialcondition ( x, y, θ, X, Y ) = (0 , , , Ω ,
0) the solution tothe (
X, Y ) subsystem is ( X ( s ) , Y ( s )) = Ω (cos( s ) , sin( s ))which gives θ ′ = ˘ κ + Ω cos( s ) which in turn recoverseq. (1) for the ( x, y ) subsystem.A purely rotating spiral wave appears motionless ina frame rotating with it. The transition to meanderingcorresponds to the Hopf bifurcation. After the bifurca-tion the spiral tip appears in the rotating frame to tracea circularly shaped path although far from the core thewave continues to appear motionless.By converting eq. (4) to a rotating coordinate sys-tem (0 , , , ,
0) becomes a fixed point with spectrum {± i ˘ κ, , µ ± i } . The eigenvalues ± i ˘ κ arise from the trans-lational symmetry of the plane and 0 arises from the ro-tational symmetry of the plane. At the Hopf bifurcationall five eigenvalues lie on the imaginary axis.Barkley showed that the type of curve traced by a spi-ral tip in the stationary frame depends on where the Hopfeigenvalues cross the imaginary axis in relation to thetranslational eigenvalues. When the translational eigen-values are between the Hopf eigenvalues the spiral tipwill follow a hypotrochoid like curve (0 < ˘ κ < < ˘ κ < κ = 1 in eq. (4)).In terms of L. monocytogenes we can interpret (
X, Y )as the projection of the cell’s translational velocity toa plane orthogonal to the cell body’s long axis and wecan interpret the oscillation of (
X, Y ) as the effect ofthe cell’s spin on its propulsion system. The long axisand the X component are parallel to the surface beingtraversed while the Y component points in the orthogonaldirection. For a cell constrained in two dimensions the Y component does not contribute to the motion. For Ω = 0the cell appears motionless in a frame rotating with it.For small Ω > L. monocytogenes motility are different but the pathsthey follow are both part of a larger two parameter fam-ily of curves. The paths followed by spiral wave tipsare organized around a first order resonant Hopf bifur-cation for which the translational and Hopf eigenvaluescoincide (˘ κ = 1 in eq. (4)). The paths followed by L.monocytogenes are organized around a zero order res-onant Hopf bifurcation for which the translational androtational eigenvalues coincide (˘ κ = 0 in eq. (4)).
4. The dynamics of L. monocytogenes motility. Spiral waves appear in excitable media when, in the statespace for the medium, the state is sufficiently close tothe appropriate attracting set. Each state in the attract-ing set corresponds to a well formed spiral wave in themedium. In many cases there is a characteristic wave-length to the limiting form of the spiral wave [39]. Insuch cases any two spiral waves in a planar homogenousisotropic medium will be congruent. For media whichsupport non-meandering spiral waves the attracting setis essentially a copy of the symmetry group SE (2). Thisis the type of attracting set the solutions to eq. (4) havewhen µ <
0. The orbits of the dynamical system in- side the attracting set are simple closed curves and thusbounded. These orbits correspond to spiral waves un-dergoing a pure rotation or
L. monocytogenes followinga circular trajectory.Aside from numerical simulations it is difficult to pre-pare excitable media so that the initial state of the sys-tem is within the attracting set, i.e. so that the mediumbegins with a well formed spiral wave which then under-goes a pure rotation. A purely rotating spiral wave onlyappears after a transient period. One way for a purelyrotating spiral wave to appear is by disrupting a circularor linear wave front with an obstacle in the medium. Thebroken end of the wave front will then curl up and overtime the shape of the wave will develop into a well formedspiral which propagates in a purely rotational manner.The disruption of a wave front by an obstacle brings thestate of the system sufficiently close to the attracting setthat it converges towards it.For homogenous isotropic excitable media which sup-port meandering spiral waves the attracting set has an-other dimension. This is the type of attracting set thesolutions to eq. (4) have when µ >
0. Each point inthe attracting set gives the position and orientation ofthe spiral wave as well as its phase within the period ofmeander. The orbits of the dynamical system inside theattracting set are bounded unless there is a resonance be-tween the rotation of the spiral wave and the oscillationof the tip in which case they are unbounded. Meander-ing spiral waves appear after a transient period once thestate of the system has been brought sufficiently close tothe attracting set. For systems at or near resonance thecore of the spiral wave will be transported across largedistances.Biological systems repeat many of the same develop-mental strategies in various contexts to form functionalpatterns. The presence of low dimensional attractingsets in complicated dynamical systems can provide sta-bility to developmental processes which are exposed tothe environment. For instance there are prokaryotes( e.g. myxobacteria) and eukaryotes ( e.g. D. discoideum )which use spiral wave dynamics to get individual cellsdispersed over a wide area to aggregate together and de-velop multicellular reproductive organs. The underlyingmechanisms by which myxobacteria and
D. discoideum move and communicate are quite different but the pres-ence of an attracting set for spiral wave dynamics canallow the aggregation process to proceed despite the va-garies of their environments.There has been a long running and continuing effort todetermine the underlying mechanism responsible for theformation of actin-based motility [5–12, 40]. The Shenoy et al. model is effective at duplicating
L. monocytogenes trajectories but it is not directly based on a physicochem-ical mechanism. Their model proceeds from general con-siderations about how the forces produced by actin poly-merization must act on the cell. In order for the cell tochange direction as it moves there must be some asymme-try in the distribution of forces exerted on the cell surface.By treating the net propulsive force as a constant paral-lel to the long axis of the cell and whose exertion pointrotates at a constant distance about the long axis themagnitude of the component of the net torque orthogo-nal to the plane of motion varies in a precisely sinusoidalfashion. In this way the cell body oscillates about itscenter of mass much like an ideal torsional spring. Withthe propulsive force always parallel to the long axis thecell moves in trajectories that alternately wind clockwiseand counter-clockwise.However this clockwork like mechanism does not sim-ply appear fully formed. For
L. monocytogenes engagedin actin-based motility the concentration of ActA (thecatalyst for actin polymerization on
L. monocytogenes )along the cell wall increases from the apical pole to thebasal pole.
L. monocytogenes cells reproduce by dividingalong a septum midway between the poles. After divi-sion each daughter cell forms a new apical pole at theseptation region and ActA concentration is redistributedalong the cell wall [14, 15].When a
L. monocytogenes cell first begins moving it en-gages in a “hopping” type motion. The density of actinbuilds up behind the basal pole until there is a suddenacceleration of the cell. The actin tail then becomes rar-efied, the cell subsequently decelerates nearly to rest, andthe cycle repeats. After several cycles the cell eventuallysettles down to a relatively constant speed [7, 16].One function served by the actin-based motility of
L.monocytogenes is the transport of bacteria from one hostcell to another without the bacteria having to leave theconfines of the host cells. This is accomplished when abacterium presses against the host’s plasma membrane tocreate a “pseudopodal projection” with the bacterium in-side. The bacterium enters a neighboring host cell whenthe pseudopodal projection is phagocytosed by the neigh-boring cell [1].During the transient period
L. monocytogenes move-ment is sensitive to obstacles in its environment but itbecomes more robust when it reaches a steady speed[16]. The presence of a low dimensional attracting setfor actin-based motility can provide stability in the de-velopment of
L. monocytogenes infectiousness. When thestate of the system is in its transient phase, away from theattracting set, obstacles in the environment can have theeffect of perturbing the system from one orbit to anotherwith a very different course. On the other hand whenthe state of the system is close to the attracting set itcan quickly return to the attracting set after an obstaclecauses a perturbation. This accounts for how
L. mono-cytogenes motility can persist as it forms a pseudopodalprojection.Moreover the attracting set for actin-based motility by
L. monocytogenes confined to two dimensions appears tobe of the same type as for spiral tip meander in two di- mensions. In both cases the points of the attractor cor-respond to the position, the orientation, and the phasein the secondary oscillation of the object that is moving.For the purposes of transport it is useful for the systemto be near a resonance. The Shenoy et al. model corre-sponds to a zero order resonance which helps account forits effectiveness although the model needs to be placed ina larger mathematical context to see this and to accountfor how actin-based motility in
L. monocytogenes arisesand persists in the presence of obstacles.
Conclusion.
Spiral waves involve the cooperative be-havior of multiple agents while a bacterium is generallythought of as a single agent. However this apparent asso-ciation of the first order resonant Hopf bifurcation to theorganized motion of multiple agents and the zero orderresonant Hopf bifurcation to the motion of a single agentis not a general principal. The movement of an individualbacterium can be regarded as the action of a single agentbut actin-based motility involves the polymerization ofactin so it can also be regarded as an organized activityinvolving many chemical agents. Inert beads coated withproteins which catalyze actin polymerization can also en-gage in actin-based motility [41, 42].Actin is an important constituent of the cytoskeletonsof eukaryotic cells and it forms locomotory structuressuch as filopodia in which actin organizes into bundlesand lamellipodia in which actin organizes into meshworks[43–49]. Numerical simulations indicate that actin canalso form spiral waves [50] and there is evidence for actinforming spiral waves inside of
D. discoideum pseudopodia[51, 52].The actin rich cytoplasm of eukaryotic cells can beseen as a type of excitable media with a propensity tolocomotory behavior which several pathogens, in additionto
L. monocytogenes , have evolved to take advantage ofin order to transport themselves. Actin-based motilityoccurs with the bacteria
Shigella flexneri [53], species of
Rickettsia bacteria [54], and vaccinia virus particles [55].The detailed mechanism of actin-based motility variesbetween these pathogens but they each involve catalyzingthe polymerization of actin, [56–58].When
L. monocytogenes bacteria are constrained tomove in two dimensions their trajectories can form a com-plicated series of coils. As with the trajectories of mean-dering spiral waves, the trajectories of
L. monocytogenes are composed of essentially congruent copies of a finitecurve repeatedly joined end to end. The form of the re-peating unit appears after a transient period and it variesbetween different occurrences of meandering spiral wavesand between different individual
L. monocytogenes . Yetcomplicated curves characteristic of both types of phe-nomena can be replicated using a two parameter familyof dynamical systems with the same type of attracting setas eq. (4). Taken together these lines of evidence supportthe idea that actin-based motility in
L. monocytogenes isorganized by the same type of low dimensional attractingset that organizes spiral tip meander.
Acknowledgment.
I would like to thank Omar Clayand Jeff Yoshimi for their comments and suggestions toimprove the [email protected] [1] Tilney L.G. and Portnoy D.A.,
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