%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% SELECT THE LAYOUT %% %% The class supports further options. %% See aipguide.pdf for details. %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[final,numberedheadings] {aipproc} % use final for the camera ready runs % use draft while you are working on the paper %%% do I need [amsmath,amssymb]? %%% \usepackage{color,graphicx}% Include figure files \layoutstyle{8x11double} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% FRONTMATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %%% Begun 3/9; Last drafts before trip, 5/21 and 6/10; to AIP format 7/19/08 \def\OMIT#1 {{}} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\eqr}[1]{(\ref{#1})} \newcommand{\half}{\frac{1}{2}} \newcommand{\la}{{\langle}} \newcommand{\ra}{{\rangle}} \newcommand {\HH} {\mathcal{H}} %% same as \Ham \newcommand {\ep}{\epsilon} \newcommand {\epdot} {\dot{\ep}} \newcommand{\xaxis}{{\bf \hat{x}}} \newcommand{\yaxis}{{\bf \hat{y}}} \newcommand{\zaxis}{{\bf \hat{z}}} \newcommand{\rr}{{\bf r}} \newcommand{\EE}{{\bf E}} \newcommand{\rcargo}{{r_{\rm cargo}}} \title{Possible mechanisms for initiating macroscopic left-right asymmetry in developing organisms} %%% *** Motor proteins (myosin, kinesin dynein) 87.16.Nn %%% *** Filaments, microtubules, their networks, and %%% supramolecular assemblies 87.16.Ka %%% *** Development and growth 87.19.lx %%% *** (Cell) Growth and division 87.17.Ee %%% * subcellular processes: Transport, incl. channels, pores, %%% lateral diffusion 87.16.dp %%% Pseudopods, lamellipods, cilia, and flagella 87.16.Qp %%% subcellular processes: analytical theories 87.16.ad %%% structure of biomolecules (incl. their chirality) 87.15.B- %%% Cytoskeleton, 87.16.Ln %%% Biol physics -- Analytical theories 87.15.ad (gen. theory and sim 87.15A-) %%% Cell locomotion, chemotaxis 87.17.Jj %%% (Higher organisms) Pattern formation: activity and anatomic 87.19.lp \classification{PACS numbers: 87.19.l, 87.16.Ka, 87.16.Nn, 87.17.Ee} \keywords {Cytoskeleton, motor proteins, actin, chirality, handedness} \author{Christopher L. Henley}{ address= {Department of Physics, Cornell University, Ithaca, New York 14853-2501, USA} } \begin{abstract} How might systematic left-right (L/R) asymmetry of the body plan originate in multicellular animals (and plants)? Somehow, the microscopic handedness of biological molecules must be brought up to macroscopic scales. Basic symmetry principles suggest that the usual ``biological'' mechanisms -- diffusion and gene regulation -- are insufficient to implement the ``right-hand rule'' defining a third body axis from the other two. Instead, on the cellular level, ``physical'' mechanisms (forces and collective dynamic states) are needed involving the long stiff fibers of the cytoskeleton. I discuss some possible scenarios; %% (C. elegans and mollusks, human brain, vertebrate internal organs) only in the case of vertebrate internal organs is the answer currently known (and even that is in dispute). \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% MAINMATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{sec:intro} The anatomy of most animals -- and many plants -- breaks left/right symmetry, the same way for all or most individuals (though this doesn't necessarily have any functional significance). The key processes at the cell or organism level -- diffusion, regulation of gene expression, perhaps elasticity -- don't distinguish left from right, so how can the developing organism ``learn'' this \cite{wolpert-brown, wood-review,levin-LR-review,wood-review-2005}~? Of course, the proteins (or other biological molecules) that constitute the cells are handed;~\footnote{ %%%%%%%%%% It should be emphasized I am {\it not} interested here in the original prebiotic symmetry-breaking which determined the molecules' handedness~\cite{frank53,avertisov-molecules,garcia-bellido}.} %%%%%%%%%%%%%%% but how can this information be brought to the macroscopic scale? Any mechanism for that, I suggest, involves {\it physics} to an unusual extent: forces and motions acting on the stiff semi-macroscopic polymers that make up each cell's ``cytoskeleton''. Furthermore, I argue that just from {\it a priori}, basic {\it symmetries}, the possible mechanisms are strongly constrained. That is why this topic seemed appropriate for a conference in honor of Landau, even though he would probably have considered the entirety of biological physics to be ``pathological''. One motivation to pursue L/R specification is ``cleaner''~\cite{palmer} than most development instabilities in that (i) it emerges out of a functionally symmetric state (if not, the original L/R specification just was earlier). Maybe, e.g., the differentiation of brain regions emerges in a similar fashion, but the way it goes can be explained by the morphological asymmetries of neighboring tissues; (ii) it is binary -- the minimum of information; (iii) mutant embryos have unambiguous visual signatures. Perhaps for these reasons, L/R experiments have surged over the past decade. The rest of the paper begins with an introduction in which I list examples of L/R asymmetry in model organisms, lay down key facts and assumptions, and classify the mechanism. The next four sections go on to tell four stories of L/R asymmetry (of which only the first is experimentally settled): cilia driving fluid flow in vertebrates (Sec.~\ref{sec:nodal-flow}), a hypothetical mechanism wherein screw processive molecular motors transport signaling molecules (Sec.~\ref{sec:screw-transport}), shearing actin arrays causing a twist in cell division as in molluscs (Sec.~\ref{sec:celldiv}), and finally rotating microtubule arrays in {\it plants} leading to a macroscopic twining as in vines (Sec.~\ref{sec:mt-plants}). \subsection{Examples} The first example is the internal organs of vertebrates~\cite{cooke-review,hirokawa-review}: heart, lungs, etc. are located asymmetrically. In humans,~\cite{mcmanus-RHLH} the frequency of mirror-reversal is $\sim~10^{-4}$. Common model species are mouse, chick, Xenopus frog, or zebrafish. A second example is the human brain:~\cite{sun-brain-review}: of course, right-hand dominance is a side effect of left-brain dominance. The congenital reversal frequency~\cite{mcmanus-RHLH} is $\sim 10^{-1}$, and %%% (after allowing for social influences) surprisingly is {\it independent} of the handedness of internal organs~\footnote{ %%%%%%%%%%%%%%%%%%%%% See references in \cite{mcmanus-RHLH}, pp. 5-6; \cite{cooke-review}, Part IV; and \cite{levin-LR-review}, Sec. 10. This independence is also confirmed in frogs~\cite{malashichev-review}.} %%%%%%%%%%%%%%%%%% so it probably has a different mechanism. (While some anatomical and functional brain asymmetries are found in other animals, the relation to human brain asymmetry~\cite{malashichev-review,halpern-LR,malashichev-book} is still unclear.) Thirdly, even the humble {\it C. elegans} nematode ``worm'' is L/R asymmetric%%%%%%%% ~\cite{wood-review,wood-review-2005,wood-orig,poole-hobert,hobert-brain-review} --- a creature so small that all $10^3$ cells are numbered by embryologists (they develop in an identical, and known, pattern), and every embryo repeats exactly the same sequence of divisions. Here, the gut twists, and a certain chemosensing neuron has functional L/R asymmetry. (Flies also have twists in their guts and genitals~\cite{ speder-drosophila-orig,hozumi-drosophila-orig,speder-noselli-review0}.) As a fourth and final animal example, mollusc shells --- say snails --- are well known to coil right-handed (reversal frequency $\sim~10^{-4}$). This has functional consequences in that mating is %%% physically awkward between snails of different handedness,~\cite{snail-species} and a snake has evolved asymmetric jaws~\cite{snail-snakes} to better crush right-handed snails. There is a parallel story of handedness in plants (Sec.~\ref{sec:mt-plants}) --- certainly in climbing plants, whose roots and shoots both spiral with a species-dependent handedness.~\footnote{ %%%%%%%%%%%%%%% Note that ``phyllotactic spirals'' in plants, responsible for Fibonacci numbers in leaf placement, in pine cones and sunfloweers --- do {\it not} have a fixed handedness.} %%%%%%%%%%%%%%% (An interesting symmetry corollary --- this would be obvious to Landau! --- is that roots growing against a vertical hard surface can deviate by a ``Hall angle'' from heading straight downwards, and they do.) Since this is biology, we do not expect universal answers; if a conjectured mechanism turns out to be wrong for one mentioned example, it might be valid for another! %% \heading{1.2 The question:} %% %% An embryo develops 2 axes: {\bf head/tail}=x, and {\bf front/back}=z, \\ %% by {\bf spont. symmetry breaking}. \\ %% (sometimes biased externally e.g. where sperm entered egg).\\ %% Then a 3rd axis (y) forms normal to them ... but \\ %% {\bf how to ensure ``right-hand rule'' ($y=x \times y$)?} %% \end{slide} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% OLD slide %% \begin{slide} %% \heading{1.3 Starting facts/assumptions/observations} %% %% \begin{itemize} %% \item[1] %% From genetic control + chirality of molecules \\ %% ({\bf not} shape of mother's womb/egg) %% \item[2] %% (Usually) L/R is spont. symmetry breaking; %% systematic bias from a small symm. breaking field.\\ %% Example: human defect Kartagener's syndrome: 50\% %% have a mirror reversal. \\ %% $\Rightarrow$ a small magnitude effect suffices %% \item[3] %% We seek the {\bf earliest} L/R asymmetry %% $\Rightarrow$ whatever makes the asymmetry, preceded it %% and was symmetrical %% \item[4] %% Expect various mechanisms, operating in the different cases, %% not a universal answer. (This is biology!). %% \end{itemize} %% \end{slide} \subsection{Question and starting assumptions} \label{sec:question-assumptions} An embryo can develop two axes by spontaneous symmetry breaking: (i) the {\it anterior/posterior} (A/P) axis, i.e. head/tail --- call this the $\xaxis$ axis; (ii) the {\it dorsal/ventral} (D/V) axis,i.e. back/front --- call this the $\zaxis$ axis. It is well understood that a combination of reaction and diffusion by chemical signals can generate this sort of pattern formation. (Here ``reaction'' includes regulation of DNA transcription and translation to proteins.) Note that in practice the symmetry breaking is often biased externally, e.g. by the point sperm entered egg. A third ($\yaxis$) axis could certainly form, normal to the others, by a spontaneous symmetry breaking. But the ``right-hand rule'' \begin{equation} \yaxis = \zaxis \times \xaxis \label{eq:rhrule} \end{equation} must be ensured: how can it be done? The key claim of this paper is that symmetry requires that {\it each of the three symbols} on the right-hand side of \eqr{eq:rhrule} has {\it a specific physical (biological) correlate} entering the mechanism. That is, there must be two kinds of preexisting polarization representing the $\xaxis$ and $\zaxis$, plus some functionally chiral element representing the ``$\times$'' in \eqr{eq:rhrule}. I mention two important assumptions. First, there is a {\it spontaneous} symmetry breaking: some robust mechanism ensures an asymmetric outcome, but that by itself would produce an equal mixture of L and R organisms. (This is evidenced experimentally, in some cases, in mutants that have such a randomization.) The uniform outcome is due to an additional {\it small} biasing field (exponentially small in the system size), just as a tiny magnetic field decides the magnetization sense of an Ising magnet cooled through its Curie temperature. The practical significance is that our mechanism need only produce a weak bias (let's arbitrarily aim for $\sim 10^{-2}$), since the assumed symmetry breaking amplifies it enormously.~\footnote{ %%%%%%%%%%%%%%%%%% Symmetry breaking may also explain the functional reason for a bias, when there is no social reason for organisms to all have the same handedness. In the absence of a bias field, occasionally an organism would be a mixture of ``left'' and ``right'' type domains, which would be a congenital defect.} %%%%%%%%%%%%%%%%%% Second, I also assume that L/R asymmetry stems from the microscopic chirality of molecules (under genetic control), and {\it not} e.g. an asymmetry produced in the egg by a right-handed mother. (This is easily refuted by the inheritance patterns; and wouldn't such a mechanism have $>10^{-4}$ error rate?) Finally, since we seek the {\it earliest} L/R asymmetry, it follows (tautologically) that whatever caused it must have been L/R symmetric.~\footnote{% %%%%%%%%%%%%%% Since all the molecules are asymmetric, more carefully we should say the cause is {\it functionally} L/R symmetric; just what that means will be clarified by examining the specific scenarios worked out in later sections.} \subsection{Classifying mechanisms} \label{sec:classify} There are three useful categories. \subsubsection{Two levels of any mechanism} Any explanation of L/R specification really requires two stories, one at the cellular level and one at the collective level. The cell level story starts from proteins --- that's where the cytoskeleton fibers come in --- and goes to properties of the whole cell, or the interaction of one cell and a neighbor. The collective story starts from the cell behavior and explains how this specifies a left and a right side in the whole embryo. For each of the following sections, I will indicate both levels; sometimes one or the other is rather trivial, but there are always the two levels. The symmetry principle applies at {\it each} level. \subsubsection{Two styles in development} In embryology,~\cite{wolpert-text} there are two general ``styles'', applying to different animal phyla. An ``early'' style applies to molluscs and {\it C. elegans}: as cells first divide, each gets a determining label, schematically like a binary string. All its descendents retain that string, while possibly adding bits that refine the specification of cell type in the mature animal. Thus, cell fates are fixed early. On the other hand, vertebrates and insects have a ``late'' style: through many cell divisions the cells are unspecified, then fates are undetermined by pattern formation within a multicellular embryo. Evidently, an essentially cell-level L/R mechanism suffices for early-style creatures, whereas a collective L/R mechanism seems to be needed in late-style creatures. \subsubsection{Two ways to represent L and R} \label{sec:ways-represent-LR} There are two ways that ``leftness'' might be represented in an embryo (Fig.~\ref{fig:LR-represent}). One is called ``positional information'':~\cite{wolpert-text} it means some chemical has a concentration $\phi(x,y,z)$ which is roughly a function of (say) $y$; three such chemicals can specify all coordinates. By sensing the concentrations of all three, a cell learns its position within the body and hence which organ it should become. (To do that, it does not necessarily need to know which {\it direction} is left, or posterior, etc.) \begin{figure} \includegraphics[width=0.90\linewidth] {LRx.eps} \caption{Two ways to represent left/right information (a). Positional information (concentration) (b). Polarization (gradient)} \label{fig:LR-represent} \end{figure} The alternate representation is ``polarization'' of cells, so that each ``knows'' which {\it direction} is left, but not where it sits along the L/R axis. We could write this as a local vector $\EE(x,y,z)$. Evidently the relation of polarization and positional information is $\EE = \nabla \phi$. Wolpert's pioneering paper~\cite{wolpert-brown} about the L/R mechanism as a symmetry problem envisaged a ``polarization'' representation. I believe that was meant only as a thought experiment, rather than a literal proposal for the mechanism; ``positional information'' seems in many cases easier to generate, as well as being the information ultimately needed to ``inform'' a developing tissue as to its fate. It is not trivial to convert one kind of L/R information to the other. Indeed, given $\phi(\rr)$, a eukaryotic cell is big enough and sophisticated enough to sense a concentration gradient between one side and the other, and develop a polarizations in response, to differentiate (in the calculus sense!). But the reverse construction --- integratiing --- cannot be done locally. To generate an imbalance in the concentration, some chemical it must get actively transported through the organism with a bias along $\EE (\rr)$. \subsection{Cartoon of the cytoskeleton} The mechanism is not of the usual ``biological'' type. That would mean transport (by diffusion or otherwise) of signaling molecules and reactions. But transport alone won't suffice for L/R, since diffusion doesn't distinguish handedness. All possible mechanisms seem to involve actual {\it forces} or {\it torques} exerted by molecular motors, which somehow structure the {\it cytoskeleton} made of stiff, semi-macroscopic fibers (that the motors run along). \OMIT{Transport creates L/R bias in concentrations only if the network geometry already had a bias} The cytoskeleton is the framework in each cell of a eukaryotic (higher) organism, built from long, stiff, directed, and helical macro-molecules (which I will call ``fibers''); there are specific kinds of motor molecules for each kind of fiber.~\cite{cytoskeleton}. The two kinds of fiber are (i) {\it microtubules} (mt), with {\it dynein} or {\it kinesin} motors on them; (ii) {\it actin} fibers, with {\it myosin} motors on them. Each family of motors contains numerous subvarieties, used by the cell for special purposes. In particular, myosin V is the main myosin variety that moves along a fiber for long distances (``is processive''); other varieties, e.g. myosin II used in muscles, only makes contractions. We note that, despite the microscopic differences between mt and actin, when abstracted to the model level they may look very similar: i.e., a physics approach may uncover (even quantify) a kind of universality. The two directions along the fibers are not symmetry-equivalent -- the growth direction is called the fiber's ``polarity''. A given kind of motor (literally) walks in a fixed sense (which we assume to be the $+$ sense for this paper). It transports ``cargoes'', which are typically chemicals in vesicles (small membrane bags attached to the motor by linking protein(s): see Fig.~\ref{fig:motors}(a)). Occasionally (by thermal fluctuation) a motor falls off its fiber, and diffuses till it reattaches to the same or another fiber \begin{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% The option [height=...] scales the picture to the given height, %% without it it would be printed at its nominal size %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \includegraphics[width=0.75\linewidth]{motors.eps} \caption{Two-legged molecular motors walk on fibers (a) carrying cargoes (b) linked to other fibers and driving a relative sliding.} \label{fig:motors} \end{figure} In place of a cargo, a motor can be linked somehow to another fiber [see Fig.~\ref{fig:motors}(b)] so as to drive the motion of one relative to the other {\it e.g.} in cell division. Fibers form networks with many crosslinks, which are often {\it dynamic}, i.e. fibers are constantly appearing, growing, shrinking, and vanishing, producing a dynamic steady state describable by statistical mechanics. \section{Nodal flow mechanism} \label{sec:nodal-flow} Here I merely review the only well-understood mechanism, a {\it late-stage} type mechanism acting in {\it vertebrates}~\cite{hirokawa-review}. \subsection{Cell-level story} We start with an approximately flat embryo that has already formed A/P and D/V axes. The following sentence describes the key and sufficient cause: watch carefully to see where each element of \eqr{eq:rhrule} gets mentioned. On its ventral side, near the ``node'' (a key place in a developing vertebrate embryo), are special cells with {\it cilia} --- moving tails that stick {\it out} (``$\zaxis$'') from surface, but {\it tilted} (``$\xaxis$'') towards the embryo's posterior end.~\cite{PNAS-nodecilia}. These cilia move {\it circularly} ($\times$), --- clockwise, looking down --- unlike regular cilia (which move back and forth). %% \includegraphics{cilia-flow.eps} \subsubsection{Root of L/R asymmetry} So, {\it why} do the node cilia move circularly? This is plausible from their structure: each cilium has 9 pairs of microtubules that run its length, and each pair is linked by dynein molecules with their head-to-foot direction oriented {\it clockwise}. The crucial L/R event is that a ring of special protein molecules assembles in the cell membrane. This serves as the template to start the microtubule pairs with dynein linkages, and the whole structure apparently grows outward ring after ring by stacking each component onto its own kind in the previous ring. In the end, then, the L/R asymmetry is not due to the inherent helicity of the microtubules, but rather the handedness of the templating complex. Plenty of physics remains to be worked out in this system, namely applying elastic theory and fluid dynamics to show why the given structure executes circularly polarized motion, and in which sense~\cite{brokaw-chiral}. \subsection{Collective level (and alternate story)} The array of cilia just described is sufficient to break L/R symmetry: symmetry allows it to drive a fluid flow L to R across the embryo (as observed). If a signaling chemical is released, the flow carries it to the L side, where it can bias the symmetry breaking. The key check is that you reverse the flow externally and get out reversed embryos --- in {\it mice}. But the relative importance of early versus late mechanisms may depend on the kind of vertebrate: early L/R asymmetries were clearly seen in {\it Xenopus} frogs~\cite{levin-ionflow-review}, due to a distinct L/R mechanism. Possibly before the first cell division, preexisting (maternal) chemicals -- in particular ion transporter proteins -- are getting asymmetrically distributed in the egg; plausibly the L/R asymmetry comes via some kind of actin/myosin mechanism like the one discussed in Sec.~\ref{sec:celldiv}, below~\cite{levin-aw}. %%% We don't yet know if that has any functional importance %%% (recall vertebrates typically fix cell fates at a later stage.) The collective level of this mechanism involves a biased transport that converts an electrical potential difference into a concentration by transport (like the ``integration'' process of Sec.~\ref{sec:ways-represent-LR}). It was speculated the collective mechanism includes a mutual feedback process (the spontaneous symmetry breaking assumed in Sec.~\ref{sec:question-assumptions}), between the electric field and serotonin concentration gradient~\cite{levin-serotonin} \OMIT{If the cilia stuck straight out, we still can get fundamentally same symmetry breaking. (Now flow is circular, and signal must be released from, say, the head end.)} \section{Asymmetric transport} \label{sec:screw-transport} This section outlines, as a pedagogical example, a completely hypothetical mechanism; it is late-stage type and (unlike the mechanisms of Sections \ref{sec:celldiv} and \ref{sec:mt-plants}) it depends on transport of signaling molecules, like typical ``biological'' mechanisms do. The key ingredient is helical motion of motor molecules on a cytoskeletal fiber. Consider for simplicity an embryo with its geometry flattened into the plane the normal to the D/V axis. Assume the dorsal and ventral sides are distinguished (there's the ``$\zaxis$ again, from \eqr{eq:rhrule}). We look for an analog of the Hall effect whereby (in the presence of a transverse magnetic field) electrons drift at a small angle from the electric field direction. Here, chemicals transported due to an assumed A/P polarization (``$\xaxis$'') actually move at an angle rotated slightly (``$\times$'') from the A/P axis, allowing them to get carried preferentially to one side of the embryo and thereby bias the assumed symmetry breaking. The upshot will be that it's not very easy to engineer late-stage asymmetry with a transport mechanism! \subsection{Cell level mechanism} This is the nontrivial level for this mechanism; again, I'll tag the three ingredients from \eqr{eq:rhrule}. Let's assume a ``cortical'' array of fibers, meaning it is just {\it under} the cell's membrane (there's the ``$\zaxis$''). Let the fibers be oriented along $\pm \xaxis$; no net polarization is assumed, so we are {\it not} yet talking about the the A/P ($\xaxis$) asymmetry. \begin{figure} \includegraphics[width=1.00\linewidth]{cargodrift-all.eps} %%% \includegraphics[width=0.30\linewidth]{cargodrift-top.eps} \caption{Mechanism of transverse drift of a ``cargo'' due to a spiraling processive motor. (a) End view (the motor is moving into the page (b) Top view (membrane is the plane of the page); the mean direction of transport is tilted by an angle $\theta_{\rm cell}$. } \label{fig:cargodrift} \end{figure} Active transport relies on motor molecules; since the fibers they move on are (microscopically) helical , their motion should (generically) be helical too [there's the ``$\times$ in \eqr{eq:rhrule}]. In fact, myosin V motors are known to spiral as they move on actin~\cite{ali-helical}. (However kinesin motors are tightly bound to the protofilaments that constitute a microtubule and those are, most often, quite straight.) Now, let's imagine our motor spirals {\it counter-clockwise} (ccw); this will pull the cargo vesicle around the fiber till it gets jammed against membrane, and from then onwards the motor will just move lengthwise: in other words, it always travels ``on the right side of the road'' [see Fig.\ref{fig:cargodrift}(a,b)]. Each time the cargo detaches, diffuses, and reattaches during its progress, the most likely outcome is to reattach to the {\it same} fiber as it is closest. If it reattaches to a different fiber, it's likelier to reattach to the fiber on its {\it right} side [Fig.\ref{fig:cargodrift}(b)] since that one is always closer than the one on the left. When we model this diffusion as confined to the plane (hence basically {\it one} dimensional), the probability of hopping one fiber to the right is $\rcargo/a$, where $a$ is the fiber spacing\cite{berg}; it is still proportional to this ratio in more realistic models. If the cargo succeeds in hopping over, its path gets shifted by $a$ to the right: hence, the {\it average} non-random shift is $\sim \rcargo$. The result is a mean transport current rotated rightwards by $\theta$ from $\pm \zaxis$, where \begin{equation} \theta_{\rm cell} \sim \rcargo / \ell \label{eq:theta_cargo} \end{equation} Here $\ell$ is the typical distance this motor goes before falling off (``processivity''), and $\rcargo$ is an effective distance of the cargo from the fiber axis. If we modeled the free diffusion interlude as a one-dimensional random walk transverse to the fiber array, the probability of reattaching to the fiber on your right is $\rcargo/d$ where $d$ is the separation; the likeliest outcome is to reattach to the same fiber, since it's closer. Since $\ell \sim 1~\mu$m and $\rcargo \sim 25$ nm, we find $\theta_{\rm cell} \sim 2.5 \times 10^{-3}$. \subsection{Collective level} This array is found only on (say) the ventral side of cells on (say) the embryo's ventral side (the ``$\zaxis$'' at organism level). Imagine the signal chemical gets released from (say) the anterior (head) end ({\it this} is the A/P or ``$\xaxis$'' asymmetry at last). If the signaling chemical has a sideways bias $\theta_{\rm cell}$ relative to the array in each cell, it's easy to see the macroscopic transport will have a bias $\theta_{\rm macro}$ of similar magnitude (see Fig.\ref{fig:drift-coll}). \begin{figure} \includegraphics[width=1.00\linewidth]{drift-collx.eps} \caption{Collective level story: if (a) the transport in each cell is rightward-biased, so is (b) the relative probabilities of transfer to a neighbor, represented in (c) as a branching ratio in a discretized model where each cell is one node of a network. Coarse-grained further to a continuum model, we end up again (d) with a bias angle. } \label{fig:drift-coll} \end{figure} \section{Cell division} \label{sec:celldiv} I now turn to an example that is not yet understood, but we know so much from experiments that we {\it should} be above to pin down the mechanism. This is (to use the categories of Sec.~\ref{sec:classify}) the early-stage, cell-level mechanism of twisting cell division (``spiral cleavage'' to the mollusc community), apparently due to an array of actin filaments. First I will review some experimental facts. For {\it molluscs}, specifically snails,~\cite{kuroda-snail-actin} after the first two divisions an embryo consists of four cells in a square. These give rise to four (smaller) daughter cells by dividing along the axis normal to the square, but break the symmetry by twisting to the right till the new cells sit on top of the furrows between the first four cells. This determins the handedness of the grown snail (one evidence is that mutants which divide at this stage with the opposite twist, also have macroscoically reversed handedness.) In {\it C. elegans} embryos, too, the embryo with four cells (planar but less symmetric) is the stage from which the ultimate handedness is determined.~% \cite{wood-review,wood-review-2005,wood-orig,poole-hobert,hobert-brain-review}. This was proven by manipulations wherein the cells get physically switched, leading to a mirror-reversal in the grown animal~\cite{wood-orig}. Furthermore, the motions of the cells in these divisions can be described by a general twisting tendency like the mollusc case~\cite{wood-review-2005}. \subsection{Organism-level story} Putting this all together, we have a mechanism for the organism level. We hypothesize a torque which always drives the same twist of two daughter cells about their axis of division. If the cells were already latently polarized along parallel axes for the subsequent division, those axes get tilted by the torque as observed. Then right after the symmetry-breaking division at the four-cell stage, some chemical signal is passed between the cells, depending on which cell neighbors. Since the neighbor relation has become asymmetric, this tells the cells which is L and which is R; that ```bit'' of information is preserved in subsequent divisions for the descendents of these cells, and is expressed functionally at a much later stage. %% \includegraphics{shibazaki-fig1.ps} %% \includegraphics{danilchik-fig2-1stcleavage.ps} \subsection{Microscopic level story?} Experimentally it was shown that the twist depends on {\it actin} but {\it not} on microtubules,~\cite{kuroda-snail-actin} somewhat surprisingly since microtubules have the more prominent role in cell division (forming the ``spindle'' between the two new cell nuclei.) The role of actin in cell division is to form the ``contractile ring'', an array of roughly parallel filaments that contract to pinch off the two cells from each other~\cite{contractile-ring,contractile-PRL}. Meanwhile, in frog eggs (under the influence of a certain drug), a twist of just this sort is observed and was shown to depend on an actin array.~\cite{danilchik-xenopus-twist}. Bundles of parallel actin fibers {\it shear} past each other, always in a {\it clockwise} sense. \OMIT{and with a fairly uniform shear} It appears myosin is responsible, rather than actin polymerization, as shown by turning off the latter with a poison. (In the case of {\it Drosophila}, a {\it late}-style mechanism which might or might not be related to this one, myosin I D is responsible~\cite{% speder-drosophila-orig,hozumi-drosophila-orig,speder-noselli-review0}.) \begin{figure} \includegraphics[width=0.8\linewidth] {array-shear.eps} \caption{ The relative shearing of two actin fibers demands a handedness relationship between the sense of their polarization(s). To drive clockwise shear, the dominant bridging bonds must be placed as shown in (a,b) sticking to the right or the left, according to the polarity of the walked-on fiber. Large arrows in the fibers indicate their polarity; in (a), the polarity of fiber A does not matter. The motor-to-actin or motor-to-motor linkages are shown as squares in (a,b); small arrows show the motion of motor relative to fiber, or fibers relative to each other. In (c), bundles are shown connected by static linkers that do not restrict the fibers' polarities; a special linker shown as $\bigodot$, binding two (oppositely oriented) motors to the membrane, can drive a shear only in the clockwise sense. } \label{fig:array-shear} \end{figure} \OMIT{ {\bf Assume:} formation of parallel actin array doesn't depend on LR. Hence it's equal mixture of both polarities. ({\sl Alternatively:} may have an overall polarity, but either sense equally likely.) %% this is part of the general principle that the mechanism that %% drives the LR is itself LR symmetric. } I do {\it not} have a satisfactory microscopic model for this case. The experiment shows the actin array's shear motion is driven by myosin, and in turn the myosin moving on one actin fiber must be linked (directly or indirectly) to another fiber, in order to make any shear (see Fig.~\ref{fig:array-shear}). So, the actin array somehow becomes organized --- either (i) driven by the myosin motors, or (ii) during the array's formation --- such that whenever a motor is attached (in any fashion) to actin fiber $A$ and walking along actin fiber $B$ (as shown in Fig.~\ref{fig:array-shear}), the polarity of fiber $B$ is towards the left as seen from the $A$ -- $B$ link.~\footnote{ %%%%%%%%%% A special case of this is where the linkage connects motors walking on {\it both} actin fibers [e.g. Fig.~\ref{fig:array-shear}(b)], as in muscle.} %%%%%%%%%% This arrangement might be produced (i) conceivably, by the spiraling of processive motors, as in the hypothetical mechanism of Sec.~\ref{sec:screw-transport}. But (ii) a more plausible mechanism would depend on the linker proteins that bind actin fibers into bundles (Fig.~\ref{fig:array-shear}(c) A ``basic'' linker would join fibers with the {\it same} polarity. But a ``special'' linker, joining fibers with the {\it opposite} polarity, would be a dimer of {\it membrane} anchored proteins oriented to hold the motors oriented as shown, so they drive relative motion only if both fibers have polarity to the {\it left} as seen from the link. \OMIT{Imagine a myosin V motor bonded to fiber A, and walking on fiber B. The motor spirals around fiber B till it comes against membrane. If the motor is on the {\it far} side of fiber B, linker strained and hence myosin falls off easier. Thus, the near-side case dominates. (To estimate this quantitatively depends on knowing the linker's elasticity.)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Plants: rotating mt array?} \label{sec:mt-plants} Finally, I turn briefly to the case of plants. Like the previous story (cell division), this involves a dynamic, cortical fiber array --- this time of microtubules (mt). %%% {\bf Dynamic fiber array \# 2, for plants} \subsection{Organism scale mechanism} Plant cells are cylindrical and elongated in the growth direction. An mt array~\cite{hashimoto-mt-review} forms around their cell walls, roughly parallel and nearly transverse to the cylinder, but with a typical helical pitch (say a pitch angle $\theta_0$ away from transverse). In turn the microtubules orient a helical array of cellulose fibers around the cells, which stiffens the cell walls. Now, the shoots (and roots) of many plant species have a consistent helical sense, particularly evident in climbing vines that twine around vertical supports. Experiments on {\it arabidopsis} confirmed the sense of the plant's macroscopic twist corresponds to that of the microtubule array's microscopic twist on the membrane: mutations that reverse the latter also reverse the former ~\cite{hashimoto}. My conjecture for how this happens comes from a paper about fungi~\cite{gamow}, where chitin plays the role of cellulose. (A related but not identical mechanism was proposed for helical twisting in chains of elongating bacteria without flagella~\cite{wolgemuth}.) Imagine a cellulose fiber of finite length $L$ anchored in the membrane (Fig.~\ref{fig:plantcell}). The cell grows by elongation but the fibril can't. Hence it feels opposing longitudinal forces at the ends, roughly proportional to $\epdot L \cos \theta$, where $\epdot$ is the fraction elongation per unit time. Since the fiber isn't longitudinal, these forces exert a twist torque on the fiber (and vice versa) proportional to $\epdot L \cos\theta \sin\theta$. Thus all the individual cells get a microscopic torque stress of the same sign, adding up to a macroscopic one on the whole shoot. Now elasticity theory tells us there can be an instability: if the centerline deviates from straight to helical, this can relieve the torque stress and decrease the energy (Such a ``twist-to-writhe'' conversion. is also responsible for supercoiling of, e.g., DNA.) \subsection{Cell level mechanism: mt arrays} \label{sec:plant-cell} Microtubule arrays get oriented in a collective process with the following rules:~\cite{mt-rules} \begin{itemize} \item{Rule (i)} The $+$ end of the mt (mostly) grows, while the $-$ end depolymerizes (but not as fast). \item{Rule (ii)} Furthermore, new mt nucleate on existing mt and grow at a specific branching angle $\sim 30^\circ$. \item{Rule (iii)} When a growing mt hits another, if the relative angle is less than $\sim 30^\circ$ it bends (this demands a linker to exert a force, since mt are rather stiff) and aligns (this process forms bundles). \item{Rule (iv)} If the angle is larger, the growing mt suffers a ``catastrophe'' meaning it {\it depolymerizes} from the $+$ end (and disappears). \end{itemize} Note that either process (iii) or (iv) tend to drive the mt towards a steady state phase in which the alignment has long range order, as found in a simulation~\cite{mt-simulation} that omits branching (ii) and bending (iii)]. Also, the collective state is symmetry-broken: within the plane of the membrane, the mt array's orientation is (so far) arbitrary. It seems difficult to engineer such a mechanism to directly orient the mt with respect to the elongation axis. Instead, I speculate there is a spontaneous (slow) {\it rotation} of the array. The overall mean orientation angle $\theta(t)$ follows the dynamics \beq \frac{d\theta}{dt} = \omega - (\alpha-\epdot) \sin 2\theta \label{eq:theta-dot} \eeq Here $\omega$ is the spontaneous rotation rate (see below for possible mechanisms). The term proportional to $\epdot$ expresses how elongation passively carries the mt to a higher angle. The consequence of \eqr{eq:theta-dot} is the angle will evolve to a steady value $\theta_0$ satisfying \beq \sin 2\theta_0 = \omega/(\alpha-\epdot). \eeq To explain the small (=nearly transverse) observed $\theta_0$, we must posit the additional transverse bias $\alpha$, which tries to pull $\theta$ back towards 0 or $180^\circ$. (Here is one speculative mechanism for $\alpha$: say there are membrane anchoring proteins that slightly bend the mt away from the membrane, so in effect the mt has a spontaneous curvature. If so, the mt has the least strain when it aligns along the direction of the membrane's maximum curvature.) Such dynamic rotation is suggested by morphologies of some plant cells that have multiple layers of cellulose fibers, each rotated relative to the one underneath~\cite{preston82}. Indeed, domains undergoing such a rotation were seen directly by video imaging in growing plant cells~\cite{mt-video} (under the influence of a drug?). The authors do not infer a particular sense, but about 70\% of the domains rotated clockwise (see Fig. 1 of Ref.~\cite{mt-video}). \begin{figure} \includegraphics[width=0.4\linewidth]{plantcell.eps} \caption{ Origin of twist torque in a plant cell. (a). The cylinder represents a cell wall. On it, cellulose microfibrils (thick segments) are templated to grow with a small tilt from the circumferential direction. (b). The cell grows by elongation. (Exaggerated in figure). The microfibrils do not elongate, but are forced to twist, so each is under tension and exerts a torque.} \label{fig:plantcell} \end{figure} So the $\omega$ term in \eqr{eq:theta-dot} is the key parameter determining L/R; what are the mechanisms for it? Recall that, by our basic symmetry arguments, the membrane {\it must} be involved in order to define a sense of rotation. The speculative ways this could happen may be classified according to the rules of microtubule growth mentioned in the preceding subsection. (a). Perhaps the microtubule-associated protein that nucleates a branch is also membrane associated [see Rule (ii), above]. (b) Perhaps the outcome of an mt-mt collision depends on which side the growing mt is impinging from [Rule (iv)], conceivably by the growing tip following a helical path of the filaments and getting pushed into or away from the membrane, but just as likely by another membrane/microtubule associated protein. Imagine the mt-rotation mechanism (b) [the one based on collisions] in more detail. If new fibers have an orientation such that they hit old fibers from the right, then they grow longer. Thus, the new fibers' orientation tends to be rotated {\it counter-clockwise} relative to the old fibers. \OMIT{Reversal mutants? Perhaps different mechanisms are simultaneously operating, thus $\omega=\omega_1 - \omega_2$. If we knock out $\omega_1$ $\Rightarrow$ in a mutation, then $\omega$ change sign, but it is not a literal reversal of the whole mechanism.} \section{Discussion and conclusions} In conclusion, I reiterate: {\it symmetry is key} to recognizing which mechanisms can possibly be responsible for L/R asymmetry. Any such mechanism must explicitly connect three ingredients: two axes, and some chiral molecule that implements the ``right hand rule''. Every story has a cell-level half and an organism half. At the cell level, a natural axis is the membrane normal, thus most of these mechanisms involved aligned ``cortical'' (adjacent to membrane) arrays of fibers. I conjecture that the mechanism is always cytoskeletal, involving microtubules or actin fibers. As for exactly how chirality enters: the most elegant ``physics'' answer would be the fiber itself, via some {\it screw mechanism} whereby motion along a long helical fiber --- walking by a motor, or microtubule collisions~\footnote{ %%%%%%%%%%%%% The stories in this paper omit a third kind of screw mechanism, namely the change in pitch of a fiber (e.g. actin) under a change of its strains, or in its chemical environment. See e.g. \cite{upmanyu}.} %%%%%%%%%%%%% gets converted into rotation around the axis. However, a ``biology'' answer, a molecule anchored in the membrane and binding the fibers too, may be more plausible. The membrane-anchoring mechanism furnishes a hint to biologists as to which proteins to focus on, in genetic or protein-expression studies aiming to discover the master L/R determining gene. On the other hand, the screw-motion hypothesis suggests that, if a processive motor is involved, the mutations which reverse (or affect) L/R determination were those that reversed (or affected) the motor's screw motion on its fiber. Twice we were led into self-organized cortical arrays of approximately parallel fibers confined to the plane adjacent to a membrane. In both cases, the conjectured cell-level mechanism did {\it not} depend on a globally defined $\xaxis$ axis, nor did it even need an ordering of the fibers' polarization axes. Instead, it used the orientation axis to define a {\it relative} rotation, so the asymmetry was manifested in a rotation or shear rate. The brain asymmetry, which supplies the very terminology (``handedness'') for this subject, is the most mysterious. Since nerve cells migrate far in the developing brain, one would hunt for an L/R asymmetry in cell locomotion. But that would occur at a later stage, in a more three-dimensional embryo, so it is less clear what the locally defined $\xaxis$ and $\zaxis$ could be. One possibility is that brain asymmetry is actually a very early mechanism, perhaps the same one discussed in Sec.~\ref{sec:celldiv}, as suggested by some left-right anomalies seen in twins~\cite{levin-birthdefects} It is left for future research to turn all these ideas into quantitative estimates, a necessary condition before any physics may be considered as completed. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% END STUFF %%%%%%%%%%%%%%% \begin{theacknowledgments} I thank Sourish Basu, Igor Segota, Michael Levin, Michael E. Fisher, and Eric D. Siggia for discussions. This work was supported by the U.S. Dept. of Energy, grant DE-FG-ER45405. \end{theacknowledgments} %%% Clifford Tabin, %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% \section*{References} \begin{thebibliography}{99} \bibitem{wolpert-brown} N. A. Brown and L. Wolpert, \emph{Development} \textbf{109}, 1 (1990). %``The development of handedness in left/right %asymmetry''. \bibitem{wood-review} W. B. Wood, \emph{Annu. Rev. Cell. Dev. Biol.} \textbf{13}, 53 (1997). %``Left-right asymmetry in animal development.'' \bibitem{levin-LR-review} M. Levin, \emph{Mech. Dev.} \textbf{122}, 3 (2005). %``Left-right asymmetry in embryonic development: %a comprehensive review''. %% lauded in one of Speder's reviews. \bibitem{wood-review-2005} W. B. Wood, \emph{PLoS Biology} \textbf{3}, e292 (2005). %``The left-right polarity puzzle: determining %embryonic handedness.'' \bibitem{frank53} F. C. Frank, \emph{Biochim. Biophys. Acta} \textbf{11}, 459 (1953). %``On spontaneous asymmetric synthesis.'' \bibitem{avertisov-molecules} V. A. Avertisov, V. I. Goldanskii, and V. A. Kuz'min. \emph{Physics Today}, July 1991, 33. %``Handedness, origin of life, and evolution''. \bibitem{garcia-bellido} A. Garc\'ia-Bellido, \emph{PNAS} \textbf{93}, 14229 ( 1996). %``Symmetries throughout organic evolution''. %% \bibitem{mislow-molecules03} %% K. Mislow, %% Collect. Czech. Chem. Commun. 68, 849-864 (2003), %% ``Absolute asymmetric synthesis: a commentary'' %% %%% review cited by M. Nilsson ArXiV 2005 %% \bibitem{saito-molecules04} %% Y. Saito and H. Hyuga, J. Phys. Soc. Jpn. 73, 33-35 (2004), %% ``Complete homochirality induced by the nonlinear autocatalysis %% and recycling.'' %% \bibitem{sendars-molecules03} %% P. G. H. Sandars, %% Orig. Life Evol. Bioph. 33, 575-587 (2003), %% ``A toy model for the generation of homochirality during polymerization.'' %% [Ou99] %% G. Ourisson and Y. Nakatani, %% Tetrahedron 55, 3183-90 (1999). %% ``Origins of cellular life: Molecular foundations and new approaches'' %% [Ze02] %% H. Zepik, E. Shavit, M. Tang, T. R. Jensen, %% K. Kjaer, G. Bolbach, L. Leiserowitz, %% I. Weissbuch, and M. Lahav, %% Science 295, 1266-69 (2002). %% ``Chiral amplification of oligopeptides in %% two-dimensional crystalline self-assemblies on water.'' %%%%%%%%%%% General %% \bibitem{supp-brueckner-comment} %% D. M. Supp, S. Steven Potter, and M. Brueckner, %% Trends in Cell Biol. 10, 41-45 (2001), %% ``Molecular motors: the driving force behind %% mammalian left-right development.'' \bibitem{palmer} A. R. Palmer, \emph{Science} \textbf{306}, 828 (Oct. 2004). %``Symmetry breaking and the evolution of development.'' %%%%%%%%%%%%%%%%%%%%%%%% %% Vertebrates %% \bibitem{belmonte04} %% A. Raya and J. C. Belmonte, %% Curr. Opin. Genet. Dev. 14, 575-581 (2004). %% ``Sequential transfer of left-right information %% during vertebrate embryo development''. %% \bibitem{belmonte06} %% A. Raya and J. C. Belmonte, %% Nat. Rev. Genet. 7, 283-293 (2006). %% ``Left-right asymmetry in the vertebrate embryo: %% from early information to higher-level integration''. \bibitem{cooke-review} J. Cooke, \emph{Biol. Rev.} \textbf{79}, 377 (2004). %``Developmental mechanism and evolutionary %origin of vertebrate left/right asymmetries'' \bibitem{hirokawa-review} N. Hirokawa, Y. Tanaka, Y. Okada, and S. Takeda \emph{Cell} \textbf{125}, 33 (Apr 7, 2006). %``Nodal Flow and the Generation of Left-Right Asymmetry.'' %% Nobutaka Hirokawa, Yosuke Tanaka, Yasushi Okada, and Sen Takeda %% pointed ou by Siggia 8/20/06. %% CLH Replied, "emailed before travel 8/06 but no reply; %% I suspect he has moved on from this topic..." %% \bibitem{taber-cardiac-review} %% L. A. Taber, %% Int. J. Dev. Biol. 50, 323-332 (2006), %% ``Biophysical mechanisms of cardiac looping.'' %% \bibitem{wilhelmi} %% H. Wilhelmi, Arch. Entwicklungmech. 48, 517-532 (1921), %% ``Experimentelle Untersuchungen uber situs inversus %% viscerum.'' %% \bibitem{tabin-review} %% C. Tabin, J. Molecular Histology 36, 317-323 (2005), %% ``Do we know anything about how left-right asymmetry %% is first established in the vertebrate embryo?''. %% \bibitem{tabin06} %% Clifford J. Tabin, Cell 127, 27-32 (Oct. 2006) %% ``The Key to Left-Right Asymmetry''. %%%%%%%%%% Brain \bibitem{mcmanus-RHLH} C. McManus, {\it Right hand, left hand: the origins of asymmetry in brains, bodies, atoms, and cultures} (Harvard Univ. Press, Cambridge MA, 2002), and references therein. \bibitem{sun-brain-review} T. Sun and C. A. Walsh, \emph{Nature Revs. Neuroscience} \textbf{7}, 655 (2006). %``Molecular approaches to brain asymmetry and handedness.'' %% \bibitem{bailly-cuif} %% L. Bailly-Cuif, Developmental Cell 12, 1-2 (2007), %% ``Breaking symmetry on time.'' %% \bibitem{belmonte} %% K. Tamura, S. Yonei-Tamura, and J. C. I. Belmonte, %% ``Molecular Basis of left-right asymmetry'', %% Devel. Growth \& Differentiation 41, 645-656 (1999). %% %%% Dev. Growth & Different. 41, 645-656 (Dec 1999) \bibitem{malashichev-review} Y. B. Malashichev and R. J. Wassersug, \emph{Bioessays} \textbf{26}, 512 (2004). %``Left and right in the amphibian world: which way to develop %and where to turn?''. \bibitem{halpern-LR} M. E. Halpern, J. O. Liang, and J. T. Gamse, \emph{Trends Neurosci.} \textbf{26}, 308 (2003). %``Leaning to the left: laterality in the zebrafish forebrain.'' \bibitem{malashichev-book} Ye. B. Malashichev and A. W. Deckel, ed., {\it Behavioral and morphological asymmetries in vertebrates} (Landes Bioscience, Georgetown TX, 2006). %% \bibitem{hatten-migration-review} %% M. E. Hatten, Science 297, 1660-1663 (2002), %% ``New directions in neuronal migration.'' %% \bibitem{klar-dynein-mitosis} %% A. Armakolas and A. J. S. Klar, %% Science 315, 100 (2007), %% ``Left-right dynein motor is implicated in selective chromatid %% segregation in mouse cells.'' %% \bibitem{gardiner-brain-tensegrity} %% J. Gardiner, J. Marc, and R. Overall, %% Frontiers in Bioscience 13, 4649-4656 (May 2008). %% ``Cytoskeletal thermal ratchets and cytoskeletal tensegrity: %% determinants of brain asymmetry and symmetry?'' %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% C elegans + molluscs \bibitem{wood-orig} W. B. Wood, \emph{Nature} \textbf{349}, 536 (1991). %``Evidence of handedness in {\it C. Elegans} embryos %for early cell interactions determining cell fates.'' \bibitem{poole-hobert} R. J. Poole and O. Hobert, \emph{Current Biology} \textbf{16}, 2279 (2006). %``Early embryonic patterning of neuronal left/right %asymmetry in {\it C. elegans}.'' %% \bibitem{poole-hobert-comment} %% B. Bowerman, %% Current Biology 16, R1039 (2006), %% ``Left-right asymmetry: making the right decision early''. %% \bibitem{heald-spindle-review} %% S. Gadde and R. Heald, %% Current Biol. 14, R797-R805 (2004), %% ``Mechanisms and molecules of the mitotic spindle.'' \bibitem{hobert-brain-review} O. Hobert, R. J. Johnston Jr, and S. Chang, \emph{Nature Revs. Neuroscience} \textbf{3}, 629 (2008). %``Left-right asymmetry in the nervous system: %the Caenorhabditis elegans model''. %%%% Drosophila \bibitem{speder-drosophila-orig} P. Sp\'eder, G. Adam, and S. Noselli, \emph{Nature} \textbf{440}, 803 (Apr. 2006). %``Type ID unconventional myosin controls %left-right asymmetry in Drosophila". \bibitem{hozumi-drosophila-orig} S. Hozumi, R. Maeda, K. Taniguchi, et al. \emph{Nature} \textbf{440}, 798 (Apr. 2006). %``An unconventional myosin in Drosophila reverses the default handedness % in visceral organs''. \bibitem{speder-noselli-review0} {P. Sp\'eder and S. Noselli, \emph{Curr. Opinion in Cell Biol.} \textbf{19}, 82 (Feb. 2007). %``Left-right asymmetry: class I myosins show the direction''. %% \bibitem{coutelis-speder-review} %% ``Left-right asymmetry in Drosophila'' %% J. B. Coutelis, A. G. Petzoldt, P. Sp\'eder, et al. %% Seminars in Cell \& Developmental Biol. 19, 252-262 (June 2008) %% \bibitem{speder-noselli-review} %% P. Sp\'eder, A. Petzoldt, M. Suzanne, and S. Noselli, %% Curr. Opinion Genetics \& Devel. 17, 351-358 (2007). %% ``Strategies to establish left/right asymmetry in %% vertebrates and invertebrates'' \bibitem{snail-species} R. Ueshima and T. Asami, \emph{Nature} \textbf{425}, 679 (2003). %``Single-gene speciation by left-right reversal: %A land-snail species of polyphyletic origin results from chirality %constraints on mating''. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% \bibitem{asami-snailmating} %% ``Evolution of Mirror Images by Sexually Asymmetric Mating Behavior %% in Hermaphroditic Snails'', %% T. Asami, R. H. Cowie, and K. Ohbayashi, %% American Naturalist 152, 225-236 (1998). %% %% Takahiro Asami; Robert H. Cowie; Kako Ohbayashi %% \bibitem{davison-nospecies} %% ``Speciation and Gene Flow between Snails of Opposite Chirality'' %% useful in guiding me to references... math. model of population %% dynamics... %% Angus Davison, Satoshi Chiba, Nicholas H. Barton, and Bryan Clarke, %% PLoS Biol 3, e282 (2005). \bibitem{snail-snakes} M. Hoso, T. Asami, and M. Hori, \emph{Biology Letters} \textbf{3}, 169 (2007). %``Right-handed snakes: convergent evolution of asymmetry for functional %specialization''. %%% DOI: 10.1098/rsbl.2006.0600 %% Snakes that eat snails with asymmetric jaw. \bibitem{cytoskeleton} {\it Mechanics of Motor Proteins and the Cytoskeleton} by Jonathon Howard (Sinauer Associates, Sunderland MA U.S.A., 2001). \bibitem{wolpert-text} {\it Principles of Development}, by L. Wolpert, J. Smith, T. Jessell, P. Lawrence, E. Robertson, and E. Meyerowitz. (3rd ed., Oxford University Press, Oxford, 2006). %% \bibitem{snail-coil-comment} %% J. Wandelt and L. M. Nagy, %% Current Biol. 14, R654-656 (2004), %% ``Left-right asymmetry: more than one way top coil a snail.'' %% \bibitem{snail-populations-rev} %% M. Schilthuizen and A. Davison, %% Naturwissenschaftern 92, 504-515 (Nov. 2005) %% %% ``The convoluted evolution of snail chirality.'' %% %% About population genetics -- is there an evolutionary reason %% %% favoring certain populations with mixed chirality? \bibitem{PNAS-nodecilia} J.H.E. Cartwright, O. Piro, and I. Tuval, \emph{PNAS} \textbf{101}, 7234 (2004). %``Fluid-dynamical basis of the embryonic development of %left-right asymmetry in vertebrates''. %%% paper on fluid dynamics of cilia. \bibitem{brokaw-chiral} C. J. Brokaw, \emph{Cell Motil. Cytoskel.} \textbf{60}, 35 (2005). %``Computer simulation of flagellar movement. IX Oscillation and symmetry %breaking in a model for short flagella and nodal cilia''; %correction at http://www.cco.caltech.edu/~brokawc/pubs.html. \bibitem{levin-ionflow-review} M. Levin, \emph{Crit. Rev. Oral Biol. Med.} \textbf{15}, 197 (2004). %``The embryonic origins of left-right asymmetry.'' %% \bibitem{levin-palmer-rev} %% M. Levin and A. R. Palmer, %% Bioessays 29, 271-287 (Mar. 2007) %% ``Left-right patterning from the inside out: widespread %% evidence for intracellular control.'' \bibitem{levin-aw} S. Aw, D. S. Adams, D. Qiu, and M. Levin, \emph{Mech. Devel.} \textbf{125}, 353 (2008). %``H,K-ATPase protein localization and Kir4.1 function reveal %concordance of three axes during early determination of %left-right asymmetry.'' \bibitem{levin-serotonin} M. Levin, G. A. Buznikov, and J. M. Lauder, \emph{Developmental Neuroscience} \textbf{28}, 171 (2006). %``Of Minds and Embryos: Left-Right Asymmetry and the %Serotenergic Controls of Pre-Neutral Morphogenesis.'' %%% a mini review, but still nearly 200 references! \bibitem{ali-helical} M. Y. Ali, \emph{Nature Struc. Biol.} \textbf{9}, 464 (2002). %``Myosin V is a left-handed spiral motor on the right-handed actin helix''. \bibitem{berg} H. C. Berg, {\it Random Walks in Biology}, second edition (Princeton University Press, 1993); see Fig. 3.2 %% \bibitem{baum-commentary} %% ``Left-right asymmetry: Actin-myosin through the looking glass'' %% B. Baum, Current Biol. 16, R502-R504 (Jul 2006). \bibitem{kuroda-snail-actin} Y. Shibazaki, M. Shimizu, and R. Kuroda, \emph{Current Biology} \textbf{14}, 1462 (2004). %``Body handedness is directed by genetically %determined cytoskeletal dynamics in the early embryo'' %% We have shown the crucial importance crucial %% of non-mirror-image cytoskeletal dynamics %% during the third cleavage of the sinistral and dextral snail embryos, %% but it is likely that the mirror-symmetry breakage occurs %% much earlier, for example, during the first cleavage. %% [I THINK THIS SPECIFIC REMARK IS DUMB, BUT SHOWS WORK TO BE DONE.] \bibitem{contractile-ring} %%%% actin role in contractile ring during cell div (cytokinesis) I. Mabuchi, \emph{J. Cell Science} \textbf{107}, 1853 (1994). %``Cleavage furrow: timing of emergence of contractile ring filaments %and establishment of the contractile ring by filament bundling %in sea urchin eggs.'' \bibitem{contractile-PRL} %%% . Contractile ring/actin physics modeling. %%%% actin role in contractile ring during cell div (cytokinesis) D. Biron, E. Alvarez-Lacalle, T. Tlusty, and E. Moses, \emph{Phys. Rev. Lett.} \textbf{95}, 098102 (2005). %``Molecular model of the contractile ring''. \bibitem{danilchik-xenopus-twist} M. V. Danilchik, E. E. Brown, and K. Riegert, \emph{Development} \textbf{133}, 4517 (2006). %``Intrinsic chiral properties of the Xenopus egg cortex: An early indicator %of left-right asymmetry?'' \bibitem{hashimoto-mt-review} T. Hashimoto and T. Kato, \emph{Curr. Opinion in Plant Biol.} \textbf{9}, 5 (2006). %``Cortical control of plant microtubules''. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% T. Hashimoto %% Curr. Opinion in Plant Biol. 6, 568-576 (2003). %% ``Dynamics and regulation of plant interphase microtubules: %% a comparative view'' \bibitem{hashimoto} I. Furutani, Y. Watanabe, R. Prieto, M. Masukawa, K. Suzuki, K. Naoi, S. Thitamedee, T. Shikanai,and T. Hashimoto, \emph{Development} \textbf{127}, 4443 (2000); %``The {\sl SPIRAL} genes are required for directional %control of cell elongation in {\it Arabidopsis thaliana}. S. Thitamadee, K. Tuchihara, and T. Hashimoto, \emph{Nature} \textbf{417}, 193 (2002). %``Microtubule basis for left-handed helical growth %in Arabidopsis;'' %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% T. Hashimoto, %%% ``Molecular genetic analysis of left-right handedness in plants'', %%% Phil. Trans. Roy. Soc. London B 357, 799 (2002). %%%%% there is abundant evidence mt are implicated \bibitem{gamow} M. P. Wold and R. I. Gamow, \emph{J. Theor. Biol.} \textbf{159}, 39 (1992). %``Fiber-composite model for helical growth in the %{\it Phycomyces} cell wall''. %%% M. P. Wold and R. Igor Gamow, \bibitem{wolgemuth} C. W. Wolgemuth, Y. F. Inclan, J. Quan, {\it et al.} S. Mukherjee, G. Oster, and M. A. R. Koehl, \emph{Physical Biol.} \textbf{2}, 189 (2005). %``How to make a spiral bacterium''. \bibitem{mt-rules} D. W. Ehrhardt, \emph{Curr. Opinion Cell Biol.} \textbf{20}, 107 (2008). %``Straighten up and fly right-microtubule dynamics and organization %of non-centrosomal arrays in higher plants''. %%% paper on rules for mt arrays %% "Live cell imaging and genetic studies are demonstrating that cortical %% microtubule arrays in plant cells are dynamic structures in which microtubule %% (MT) bundles play a key role in creating array organization and function. %% Steps important for creating and organizing these arrays include recruitment %% of nucleation complexes to the cell cortex and to the lattices of previously %% established MTs, association of newly created MTs to the cell cortex, release %% of MTs from sites of nucleation, transport of released MTs by polymer %% treadmilling, and subsequent interactions between treadmilling MTs. The results %% of MT interactions include induced catastrophe, severing, and the capture and %% reorientation of growing polymer ends by bundling interactions. Together, %% these properties predict a capacity for self-ordering that is likely to play %% an important role in establishing the parallel organization of the arrays. \bibitem{mt-simulation} V. A. Baulin, C. M. Marques, and F. Thalmann, \emph{Biophys. Chemistry} \textbf{128}, 231 (2007). %``Collision induced spatial organization of microtubules''. %%% Strasbourg mtubule simulation, c 2007 \bibitem{preston82} R. D. Preston, \emph{Planta} \textbf{155}, 356 (1982). %``The case for multinet growth in growing walls of plant cells''. \bibitem{mt-video} J. Chan, G. Calder, S. Fox, and C. Lloyd, \emph{Nature Cell Biol.} \textbf{9}, 171 (2007). %``Cortical microtubule arrays undergo rotary movements %in {\it Arabidopsis} hypocotyl epidermal cells''. %%% Video of mt, c 2007 \bibitem{upmanyu} M. Upmanyu, H. L. Wang, H. Y. Yang, and R. Mahajan, \emph{J. R. Soc. Interface} \textbf{5}, 303 (2008). %``Strain-dependent twist-stretch elasticity in %chiral filaments''. \bibitem{levin-birthdefects} M. Levin, \emph{Birth Defects Research (C)} \textbf{78}, 191 (2006). %"Is the early left-right axis like a plant, a kidney, %or a neuron? The integration of physiological signals in %embyronic asymmetry". \end{thebibliography} \end{document}