Transition to superparamagnetism in chains of magnetosome crystals



[1] Magnetotactic bacteria use chains of magnetic crystals to orient them in the Earth's field. Magnetic measurements show that these chains are in a single-domain state with all the moments pointing along the chain axis. Yet many of the crystals in the chains fall in either the multidomain (MD) or the superparamagnetic (SP) size range for isolated crystals. Magnetostatic coupling between crystals keeps the magnetization uniform and prevents thermally assisted transitions between states. The SP critical size for a chain of magnetite crystals is calculated using a new algorithm. The network of stable states and transition states connecting them is determined using a homotopy continuation method. This determines the energy barriers between stable states, and the critical size can then be calculated using a master equation. As the number of crystals in the chain increases, the SP critical volume approaches a limit that is nearly independent of the shape of the crystals. The cube root of this volume is about 10 nm. Most magnetosome crystal sizes are well above this limit. About half of the magnetosome crystals found in sediments would be MD in isolation. However, there is also a population of cubo-octahedral crystals, a large fraction of which would be SP in isolation. Such crystals are also formed in fresh water by bacteria in the genus Magnetospirillum. The difference in size and shape between the populations of isometric and nonisometric crystals may be related to redox conditions and the choice of magnetoaerotaxis mechanism.

1. Introduction

[2] Magnetotactic bacteria form one of the few groups of micro-organisms to leave a fossil record in the form of chains of magnetic crystals. These magnetofossils reflect conditions that favor growth of magnetotactic bacteria and may therefore serve as proxies for ancient biogeochemistry [Hesse, 1994] as well as more recent environmental processes [Egli, 2004]. As magnetofossils they are useful to the extent that their physical and magnetic properties are the result of natural selection in the geomagnetic field.

[3] Many strains of magnetotactic bacteria are concentrated within narrow redox zones in marine or lacustrine environments [Faivre and Schüler, 2008]. Their magnetic moments orient them in the Earth's magnetic field, countering the randomizing effect of Brownian rotation. Generally the moments are enough for an average of 90% or greater alignment with the field [Kalmijn, 1981].

[4] Iron, like other elements, is often in short supply, so magnetotactic bacteria need magnetic crystals that generate a maximum torque per unit iron. This is achieved in the single-domain (SD) state, in which the magnetization in a small field is saturated in a direction determined by its magnetic anisotropy. There is a characteristic size range for the single-domain state. Smaller crystals are superparamagnetic (SP), having a moment that is destabilized by thermal fluctuations; larger crystals are multidomain (MD), having a stable but small moment. There is at least one strain of bacterium that collects SP magnetite crystals internally [Glasauer et al., 2002], but there is abundant evidence [e.g., Moskowitz et al., 1993; Dunin-Borkowski et al., 1998; Penninga et al., 1995] that in most strains of magnetotactic bacteria the crystals are SD and all the crystal moments are aligned with the chain axis.

[5] The size and shape of magnetosome crystals are typically narrowly defined and characteristic of the bacterial strain. There are a variety of shapes, including cubo-octahedral, elongated prisms, arrowhead, tooth- and bullet-shaped [Spring and Bazylinski, 2006]. Most sizes range from about 30 nm to 120 nm. The narrow size distribution should be a useful biomarker for magnetosome crystals. An objective size criterion would be the theoretical limits for the single-domain state, which depend on the crystal shape. Unfortunately, the single-domain size range for cubo-octahedra is 47 nm < L < 55 nm (section 2).

[6] These limits do not take into account another property of magnetosome crystals: their arrangement in chains. The magnetostatic interactions between crystals stabilizes the magnetization. In this article I will calculate the effect of magnetostatic interactions on the superparamagnetic critical size. I will concentrate on strains of bacteria that grow the crystals in chains of vesicles called magnetosomes. Other arrangements are possible. For example, Magnetobacterium bavaricum grows them in long linear bundles [Hanzlik et al., 2002].

2. Previous Work

[7] Early works such as Frankel et al. [1979] cite the critical size calculations of Butler and Banerjee [1975] as support for the hypothesis that magnetosome crystals are single domain. The upper critical size is the size above which a MD state has lower energy than an SD state. However, the magnetization does not have to be in the lowest-energy state. As the crystal size increases, nonuniform states appear first on minor loops in the hysteresis [Newell and Merrill, 2000a] while the saturation remanent state remains SD. Using a numerical micromagnetic model for cuboids, Fabian et al. [1996] calculate an upper critical size for a metastable SD remanent state, and Newell and Merrill [1999] calculate it rigorously for spheroidal crystals using nucleation theory. However, the metastable size range is negligible for cubes and spheres. The upper critical size for a cube with moment along a 〈111〉 easy axis is 55 nm. Single-crystal critical sizes have also been calculated for other shapes using numerical micromagnetic models [e.g., Witt et al., 2005]. For small length-to-width ratios, the differences between these models and Butler and Banerjee [1975] are modest.

[8] The superparamagnetic critical size obtained by Butler and Banerjee [1975] combines two calculations. Both use the Néel [1955] expression for the relaxation rate of a single-domain crystal:

equation image

where ν0 is a frequency factor (which they assumed was 109 s−1), T the temperature, kB the Boltzmann factor, V the volume of the crystal, and ΔgV the energy barrier between states of positive and negative saturation. The critical volume Vb is the volume for which 1/ν is equal to the time scale of interest. The energy is determined by the magnetic anisotropy of the crystal. For elongated crystals, Butler and Banerjee [1975] calculate the energy barrier using only the shape-dependent magnetostatic anisotropy. Their prediction of the critical size is fairly good for width-to-length ratios less than about 0.9, but goes to infinity as the ratio approaches unity. For cubes, they use Δg = ∣K1∣/12, where K1 is the (negative) cubic magnetocrystalline anisotropy parameter. This gives the correct critical size for cubes. A micromagnetic calculation by Winklhofer et al. [1997] agrees well with Butler and Banerjee [1975] except for cubes. Newell and Merrill [2000a, 2000b] derive accurate analytical expressions for relaxation rates at all aspect ratios, taking into account the mixture of cubic and uniaxial anisotropy. From these expressions the critical size can be calculated. This size, if defined as the cube root of the volume, is 47 nm for any shape with cubic symmetry.

[9] The above improvements in critical size calculations are still not adequate for magnetosome crystals because they do not take into account the magnetostatic interactions between magnetosome crystals. Muxworthy and Williams [2006] estimate the upper critical size for chains of three crystals using a numerical micromagnetic model. They start the model with a single-domain initial guess and find the largest size at which the model converges on a single-domain state. They find that interactions increase this size substantially at all aspect ratios.

[10] Muxworthy and Williams [2009] also attempt to calculate the SP critical size for chains using a modified single-domain theory. If a single-domain crystal has uniaxial anisotropy then the normalized energy barrier in equation (1) is

equation image

where μ0 is the permeability of free space, Ms the saturation magnetization and HK the coercivity in a field parallel to the easy axis (often called the microcoercivity). Muxworthy and Williams [2009] start with a more complex model, but in the end they use the above expression to calculate energy barriers for chains (with HK now called Hc). HK is calculated using a micromagnetic simulation of the magnetic hysteresis in a field parallel to the chain axis.

[11] This approach should give the correct trends in critical sizes because stronger interactions increase both the coercivity and the energy barriers in zero field. Quantitatively, however, it only works for uniaxial SD crystals. A single crystal with cubic anisotropy has a microcoercivity HK = 4∣K1∣/3 μ0Ms [Newell and Merrill, 1999], which inserted in equation (2) gives Δg = (2/3)∣K1∣. Since the true value is Δg = ∣K1∣/12, the Muxworthy and Williams [2009] model underestimates the critical volume by a factor of 8.

[12] The superparamagnetic critical size has not been calculated for interacting crystals, but it can be determined using analytical expressions for the thermal decay of the moment of a chain of crystals [Chen et al., 1992; Klik et al., 1994]. If the magnetostatic interaction is strong (relative to the intrinsic anisotropy of the crystals), there are only two stable states corresponding to both moments up or both down. The moments reverse by the symmetric fanning mechanism, as defined by Jacobs and Bean [1955]: one moment rotating clockwise, the other anticlockwise at the same rate. Below a critical ratio of interaction strength to anisotropy, the intermediate states (up, down) and (down, up) are stable and the reversal mode is something that Chen et al. [1992] call “asymmetric fanning” because the moments rotate in opposite directions but by different amounts. However, the name of this mode is misleading because only one of the moments ends up reversing. The other moment also rotates away from the axis but returns to its initial state. In general, the thermally assisted reversal mode in zero field is not the same as the switching mode at H = HK, another reason why a modified single-domain theory cannot accurately predict the SP critical size.

[13] Hendriksen et al. [1994] model thermally assisted reversal in chains of interacting ferromagnets. They find that the preferred mode for strong interactions is one in which the reversal starts at one end of the chain and propagates to the other. The method they used was to force the moment to gradually change from positive to negative saturation, minimizing the energy at each step. Lyberatos and Chantrell [1995] question the reliability of this method, noting that inconsistent results have been obtained using this method. They use a ridge optimization method to obtain reversal modes for chains of three particles.

3. Methods

[14] The general approach I will use to calculate critical sizes is as follows:

[15] 1. Use Lagrange multipliers to express the equations for stationary points in the energy as polynomial equations. Use a sophisticated polynomial solver to find all the roots of the polynomial system.

[16] 2. From the stationary points select the stable states and transition states. Determine how they are connected by following the steepest descent lines departing from the saddle points.

[17] 3. Use the differences in energy between transition states and stable states to determine the relaxation rates between stable states.

[18] 4. Formulate a master equation for the time evolution of the probability density of each state. Solve it for the time evolution of the total moment.

[19] 5. Find the volume that results in a 1/e decay of the moment over the reference time scale. This is the SP critical size for that aspect ratio.

[20] To make the calculations easier, I assume that the crystals are identical cuboids (rectangular prisms), evenly spaced with long axes parallel to the axis of the chain and to the 〈111〉 easy axis in each crystal. Some of these assumptions are more accurate than others. In most strains of bacteria the long axis of each crystal is always the 〈111〉 axis. Although in many images of magnetosome chains the chains have kinks, this is probably the result of shrinkage during preparation [Shcherbakov et al., 1997], so it is reasonable to assume the chains are straight. The cuboidal shape with square cross section is convenient for calculating magnetostatic coupling. Although no natural magnetosome crystal has this shape, the approximation does not introduce much error (see section 3.1.2).

[21] The most critical assumption is that all the crystals are the same size. Through most of a chain, the crystals tend to have very similar sizes and shapes. The exception is at the ends of the chain, where the crystals are often smaller, probably because they are still growing [Mann and Frankel, 1989]. By forcing all the particles to have the same size, I am modeling a process in which all the crystals are growing at the same rate until they are large enough for the magnetization to be stable. This is not what actually happens. Usually a bacterium inherits part of a chain from its parent and starts growing new crystals at the ends. However, while the bacterium is waiting for new crystals to finish growing, the starting chain must have a stable moment, and most of the crystals in the chain have similar sizes. Thus, the critical size for identical crystals establishes a minimum size for most of the crystals.

3.1. Energy and Equilibrium Equations

[22] Stable magnetic states in a ferromagnet are minima of the Gibbs free energy G under the constraint that the magnetization at any point in the body has fixed magnitude and can only rotate [Brown, 1978]. This implies that the moment of a uniformly magnetized crystal with volume V is VMsequation imagei, where Ms is the saturation magnetization and equation image = (α, β, γ) is a unit vector with α2 + β2 + γ2 = 1. In this section, I develop an expression for the free energy, taking into account symmetries to simplify the expression.

3.1.1. Energy Terms

[23] The energy of a chain of uniformly magnetized magnetosome crystals has two parts: the magnetostatic energy and the magnetocrystalline anisotropy energy. Suppose there are N crystals with identical dimensions Δr = (W, W, L), positions ri and unit vectors equation imagei = (αi, βi, γi). The demagnetizing energy is [Newell et al., 1993]

equation image

where V = W2L is the volume and μ0 is the permeability of free space. The generalized demagnetizing tensor equation imager, rirj) is a dimensionless coupling term that depends on the geometry of the chain: the shapes of the magnetosome crystals (represented by Δr), their orientation and the positions ri of their centers relative to some arbitrary reference point. In the rest of this article, the crystal dimensions are implied in reference to equation image.

[24] Magnetite has a cubic anisotropy that adds fourth-order terms to the expression for the energy. Most magnetosome crystals are oriented so the 〈111〉 axis is parallel to the chain and is the axis of elongation. If the cubic anisotropy were large enough, it would greatly increase the complexity of the energy surface. Fortunately, if the cubic anisotropy is small enough it can be replaced by an effective uniaxial anisotropy.

[25] For a crystal elongated in a 〈111〉 direction, Newell [2006b] showed that for sufficient elongation the energy barrier is effectively uniaxial with energy barrier (Ku(d)K1/12), where Ku(d) represents the shape anisotropy and K1 is the cubic anisotropy parameter. The criterion for sufficient elongation is Ku(d) > 0.2229∣K1∣, which only requires a length-to-width ratio of 1.3. This energy barrier is also correct for K1 = 0. Since the critical size changes smoothly in between, this expression is also quite accurate for intermediate ratios.

3.1.2. Simplifications

[26] Let the chain axis be in the z direction and the crystals be indexed by their position in the chain, with i = 1 at one end and i = N at the other end. If the distance between adjacent magnetosome centers is Δz, all the demagnetizing tensors have the form equation image((ijz) (where the dependence on Δr is implicit).

[27] Given this geometry, the principal axes of the tensors coincide with the coordinate system. The off-diagonal components are zero because contributions from opposing faces cancel out [Newell et al., 1993]. The self-demagnetizing tensors (i = j) have trace 1 while the coupling tensors (ij) have trace 0. Finally, I will further simplify the problem by assuming that the magnetosome crystals have square cross sections, so Nxx(r) = Nyy(r). Then

equation image

and for ij,

equation image

[28] The difference Nd ≡ (Nxx(0) − Nzz(0)) is the usual demagnetizing factor for an isolated SD crystal and is nonnegative for lengths L and widths W such that LW. If we define

equation image
equation image

the total normalized energy is (aside from a constant that can be ignored)

equation image

Since Nd(−r) = Nd(r) [Newell et al., 1993], ki(d) = ki(d). Note that, under the constraints of this problem, the ki are functions of the aspect ratio q = L/W. In this article k1 varies between about 0.02 and 10, and ki is monotonically decreasing with increasing i. The ki are calculated for cuboids, but in the single-domain approximation details of the shape aside from aspect ratio do not matter much. For example, the self-demagnetizing factor (i = 0) for a spheroid [Chikazumi, 1997] is at most 20% larger than the same factor for a cuboid [Rhodes and Rowlands, 1954; Newell et al., 1993].

3.1.3. Planarity Constraint

[29] Because of the rotational symmetry of the energy, a given solution can be rotated about the chain axis by an arbitrary angle to get another solution. To make the solution unique, the longitudinal angle of at least one moment must be specified. However, we can do better than that. Each coupling term has the form kij(αiαj + βiβj − 2 γiγj). If spherical polar coordinates are defined such that θi ∈ [0, π) is the angle of moment i with respect to the chain axis and ϕi ∈ [0, 2π) is the longitudinal angle, the coupling term is gij = kij(cos(ϕi − ϕj) sinθisinθj − 2cosθicosθj). The derivatives of this term with respect to ϕi and ϕj are zero if ϕi = ϕj or ϕi = ϕj + π. If all the moments are in the same plane, then any state satisfying ∂g/∂θi = 0 for all i is an equilibrium state because all the derivatives with respect to ϕi are also zero.

[30] Now suppose that all the ϕi are in the same plane and have values 0 or π. Each nearest-neighbor term has a Hessian {∂2gi,i+1/∂ϕi∂ϕi+1} with determinant 1, consistent with this term being a minimum with respect to ϕi and ϕi+1. Since that is true for all indices i, the planar solutions are minima with respect to out-of-plane rotations. This argument only shows that the solutions are local minima. It is possible that there are nonplanar energy minima, but it is reasonable to assume that the moments are in the XZ plane. The reduced energy is now

equation image

3.2. Equilibrium States

[31] Energy minima, maxima and saddle points are all equilibrium states of the system. The constraint αi2 + βi2 + γi2 = 1 is commonly incorporated by expressing the energy in terms of spherical polar coordinates, in which case the conditions for equilibrium are that all first derivatives of the energy with respect to these angles are zero. An alternative approach is to add a Lagrange multiplier for each magnetization vector. This is not commonly done, but I will show that it is advantageous for this problem. Then I will show how to find the stable states and saddle points, and determine how the saddle points are connected to stable states.

3.2.1. Finding the States

[32] If we restrict the moments to the XZ plane, an isolated crystal with uniaxial anisotropy has two stable states with moments in the ±z directions and two energy maxima with moments in the ±x directions, for a total of 4 equilibrium states. Thus, N noninteracting crystals have 2N stable states and 4N equilibrium states. That is over a million states for 10 crystals! All these states are easily listed because they are combinations of the single-crystal states. However, if magnetostatic interactions are introduced, it becomes much harder to solve for these states. Normally in micromagnetic problems one provides an initial guess for a solution and looks for the nearest solution, but such a method is not useful for finding all the solutions. In general, we do not even know how many solutions there are.

[33] Fortunately, all the solutions of a polynomial system can be located using homotopy continuation [Verschelde, 1999]. The equilibrium equations for the magnetic moments can be expressed as polynomial equations if Lagrange multipliers are used. If we define

equation image

then the system of equations

equation image

are polynomials in the 3N variables αi, γi and λi. This can be represented as F(x), where x is a vector containing all the αi, γi and λi.

[34] The algorithm for finding all the roots of F(x) has four parts [Verschelde, 1999]: (1) Scale the equations and reduce their degree, if possible. (2) Estimate the number of solutions and construct a starting polynomial system G(x) with the same number of (easily determined) roots. (3) Use homotopy continuation to follow paths from the roots of G(x) to the roots of F(x). (4) Refine the roots and validate them.

[35] A crude estimate of the maximum number of solutions of these equations is the total degree of the system. This is the product of the degrees of the individual polynomials. In this case all the polynomials are second-order and there are three equations per crystal, so the total degree is 3N. However, many of these solutions are unphysical “solutions at infinity.” Root-counting methods that exploit the structure of the polynomial system give much better estimates of the number [Verschelde, 1999]. They also construct a starting polynomial G(x) with random coefficients that has the same structure.

[36] Given F(x) and G(x), a function

equation image

is constructed. As t varies from 0 to 1, the polynomial system transforms smoothly from the start system to the target system. A predictor-corrector method is used to track each root as a function of t. The solutions are tracked in a complex projective space [Wise et al., 2000], which shortens paths and eliminates almost all divergent paths. Moreover, in this space the paths do not turn back as t goes from 0 to 1 [Verschelde, 1999].

[37] Finally, the approximate roots are refined and validated. The result is that all the solutions of F are found with their multiplicities.

3.2.2. Classifying and Connecting the States

[38] Once the equilibrium states are known, they must be classified as stable states and saddle points, and other equilibria such as energy maxima. Since the moments are restricted to unit circles (αi2 + γi2 = 1), we can express the energy in terms of a circular angle ψ such that α = sinψ and γ = cosψ. The eigenvalues of the Hessian of the reduced energy, Hij = ∂2g/∂ψiψj, determine the stability. If they are all positive, the state is stable. If they are all positive, the state is an energy maximum. Otherwise it is a saddle point. Only first-order saddle points, those for which the Hessian has just one negative eigenvalue, are on the minimum energy path between stable states [Mezey, 1977]. Such a saddle point is a maximum in only one direction, and the steepest descent path down either side of this maximum leads to a stable state. I will refer to first-order saddle points as transition states.

[39] For a rough idea of how the transition states are connected to the stable states, one can plot the moments of the states against the energies (Figure 1). Whatever the path between one state and another, it must pass through all intermediate values of the moment; and a saddle point can only connect two stable states if it has a higher energy than either.

Figure 1.

Equilibrium states as a function of total moment along the chain axis (divided by the single-crystal moment) for three crystals with L = 2W and no gap between them.

[40] Even with the above considerations, there are occasions when the correct connection between stable states is not obvious. The connection can then be determined by going downhill from the transition state. The eigenvector associated with the negative eigenvalue of the Hessian defines an axis along which a downhill gradient leads away from the transition state in both directions. Each gradient leads to a unique energy minimum. I use a two-step method for finding the energy minima. In the first step, I follow the energy gradient down from the transition state by solving the differential equation equation imagei = ∂g/∂ψi, where the dot denotes a time derivative. The transition state itself cannot be used as a starting point because all the derivatives are zero, so I add a perturbation that is a small positive or negative multiple of the eigenvector. Once this equation gets sufficiently close to the energy minimum, an energy minimization routine can take over for more efficient convergence on the minimum. I found the correct balance between the two stages by trial and error. The first stage was implemented using the MATLAB ODE solver ode45 [Shampine and Reichelt, 1997] and the second by fminunc, an implementation of the BFGS Quasi-Newton method with a cubic line search procedure [Nocedal and Wright, 2006].

[41] An example of the network of paths between energy minima is shown in Figure 2. There are only two crystals, so the paths can be plotted on the energy surface. If there are more than two crystals, one must be content with a profile of energy against total moment as in Figure 1. In general, these paths are only a guide to the connections and not the actual path taken by moments during a thermally assisted transition. This path is determined by a combination of precession, damping (which makes the energy decrease over time) and thermal fluctuations [Brown, 1979]. In the limit of high damping the moments would follow the steepest descent path down from a transition state, but the uphill path would be a rare event in which thermal fluctuations keep raising the energy and therefore would be much more irregular. Actual ferromagnets have low damping, so their dynamics would be dominated by precession.

Figure 2.

(a) Energy landscape for two interacting crystals with L = 2W (based on results by Chen et al. [1992]). The angles ψ1 and ψ2 are the angles with respect to the chain axis. The lowest-energy states are those with moments aligned: (ψ1, ψ2) = (0, 0) or (π, π). Reversal of the moment occurs by way of the other two stable states (0, π) and (π, 0). Transition states are represented by stars. (b) A schematic of the network. The relaxation rate across a single barrier is ν1 or ν2, and there are two connections between each pair of stable states.

3.3. Master Equation

[42] The moments in a chain of ferromagnets are constantly fluctuating. If there are N moments, the collective state can be represented by a vector in a 2N-dimensional space. However, near the SP critical size this vector is rarely far away from a direction of minimum energy. For such a system the high-energy barrier approximation [Brown, 1979] is appropriate. In this approximation, a local equilibrium is established near each minimum. The local distribution of moments is determined by the Boltzmann distribution, and this distribution is a very steep function of distance from the minimum. Transitions between states are rare. The energy surface is also steep near transition states, so during a transition the moment is very likely to pass near the transition state. It follows that transitions only occur between states that are directly connected by a transition state.

[43] If a pair of states i and j is directly connected, there is a transition rate νij from state i to state j and another νji from state j to state i. If the states are not directly connected, νij = νji = 0. If the normalized energies of the connected states are gi and gj, and that of the transition state is gsij, then

equation image

where kB is the Boltzmann constant. The prefactor ν0 is determined by the detailed shape of the energy surface and the level of damping [Brown, 1979; Newell, 2006a], but a value of ν0 = 109Hz is accurate enough for blocking calculations.

[44] Let ni(V, t) be the probability of state i at time t for particles of volume V. The probability depends on a lot of other variables such as the temperature T, but the goal is to solve for a critical volume V with all the other variables fixed. Let mi be its moment in the z direction. The time evolution of the system is given by

equation image

This can be converted [Brown, 1979] to a master equation for n:

equation image

where Wij = νij for ij, Wii = −∑jνji. Formally, the solution of this equation is

equation image

where the exponential is interpreted as a matrix exponential [e.g., Horn and Johnson, 1994, p. 204]. Such solutions have multiple decay times, not all of which affect the moment of the chain [Newell, 2006a, 2006b]. However, the point of Néel's theory is that, for sizes slightly larger than the critical size, the moment does not change significantly over the time scale of interest, tb, while for slightly smaller sizes the moment rapidly goes to zero. It is therefore better to look at the time dependence of the moment:

equation image

[45] The Néel relaxation time is the time at which the moment has decayed to 1/e of the original moment:

equation image

To solve for the SP critical volume, substitute t = tb in the above equation to get an implicit equation for V. The solution of this equation is the critical volume. In practice, I express m as a function of L = V1/3 because m(V, t) is an extremely sensitive function of volume. For the same reason, the initial guess must be quite accurate to avoid overflow or underflow. One way to obtain a good initial guess is to use the solution for a slightly larger or smaller aspect ratio. For an elongated crystal, the noninteracting limit can be used. To check that the critical volume is not sensitive to the fraction of the original moment, I also calculate it for m(V, t) = 0.01 m(V, 0).

[46] Most studies of magnetosome crystal size include SP critical size curves for time scales of 100 s and 4.5 Ga. These were used by Butler and Banerjee [1975] to represent a laboratory time scale and the upper limit of paleomagnetic time scales. The latter is certainly not relevant to magnetotactic bacteria, while the former is too short. A magnetotactic bacterium that loses its magnetic moment in 100 s will not thrive. The moment should last at least long enough for cell division to occur. The chain is divided between the offspring and must provide a moment to orient new magnetosome crystals when they are grown. The cell division time depends on nutrient availability, but a reasonable estimate is 105 s, or about 1 day. This estimate can be off by orders of magnitude without much effect on the SP critical size.

3.4. Lowest-Energy Paths

[47] I use two approaches to calculating critical sizes. The first is find all the stable states and determine the network of paths connecting them across transition states. This is the most accurate approach, but the network can be very complicated. Usually the decay of the moment is dominated by the lowest-energy path between the states of positive and negative saturation. My alternate approach is to find a lowest-energy path and use it to calculate the critical size. This requires some adjustment. For each stable state with moments m(zi) another can be obtained by reflection about a plane through the center of the chain: m′(zi) = m(−zi), where the origin is equidistant from the ends of the chain. This transformation maps the two saturated states onto each other, but maps all other states on minimum-energy paths onto other states with the same energy and total moment. Furthermore, each transition has an image under a 180° rotation about the chain axis. Using these symmetries I use one path to build up the complete minimum-energy network. For example, in Figure 2b, the route 1 → 2 → 4 can be used to construct 1 → 3 → 4. The transition matrix for this system is

equation image

If this system begins in the state of positive saturation, the initial (normalized) moment is 2 and the time evolution of the moment is

equation image

This is the same time evolution that one would get in a two-state system with barrier rate 2ν1. In this section, I discuss the states on the lowest-energy path, but calculate the critical sizes using the full network.

4. Results

[48] All the equilibrium states were obtained using a MATLAB interface [Guan and Verschelde, 2008] to PHCpack [Verschelde, 1999]. The number of solutions is generally 4N, many of them complex. As the aspect ratio decreases the relative strength of interactions increases and the proportion of real solutions declines. For example, when N = 3, there are 64 real solutions for large aspect ratios and 16 for small aspect ratios.

[49] If the crystals do not interact, the stable states are all the combinations of up and down parallel to the chain axis, for a total of 2N states, all of which have similar energies. As the aspect ratio decreases, the energies spread out (Figure 3a). An increasing number of these states become unstable until only the two saturated states remain (Figure 3b). Stable states that have off-axis moments are rare.

Figure 3.

(a) Energies of equilibrium states for a chain of three crystals and (b) number of each type. The colors in the legend apply to Figures 3a and 3b. The numbers of states in Figure 3b are larger than the number of points in Figure 3a because many of the states are degenerate.

[50] The magnetic coupling between crystals depends on the distance between them. The spacing of crystals varies considerably between bacterial strains. For example, the more elongated crystals in the marine vibrio, strain MV-1, are separated by gaps almost as large as the crystals [Dunin-Borkowski et al., 1998], while those in marine spirillium strain MV-4 are almost touching each other [Bazylinski and Frankel, 2004]. In this section I will look at an extreme case, that of crystals touching each other, then look at the effect of crystal spacing where it matters most: crystals with no elongation.

4.1. Three Crystals

[51] As the aspect ratio q changes, so do the number of stable states and transition states (Figure 3). For this number of crystals and more, most of the changes in topology occur for aspect ratios less than 2 and most do not affect the minimum energy path.

[52] A chain of three crystals has three aspect ratio intervals with different minimum energy paths. In the first two intervals, the stable states are the two saturation states with moments all up or all down (Figure 4). For 1 ≤ q ≤ 1.16, the transition occurs by the symmetric fanning mode in which all the moments rotate equally but in alternate directions (Figure 4a). The transition state has moments in alternating x directions. Note that I use ≤ signs because my calculations are for increments of 0.02 in the aspect ratio.

Figure 4.

Minimum energy path for three crystals, L = W and no gap between crystals. The horizontal axis is the total moment in the z direction, and the vertical axis is the energy (both arbitrary units). Schematics of the transitions from negative saturation to the transition state for (a) q = 1 and (b) q = 1.3. The initial moments are unit vectors indicated by the solid red lines. During the transition the moments follow the solid blue arcs to the transition states (triangles) and then carry on along the dashed arcs. The final moments are represented by dashed red lines with circles at the end. The transition paths are represented by a light blue as a reminder that they are only guides, not the actual path followed by the magnetization (section 3.2.2).

[53] In the next interval, 1.18 ≤ q ≤ 1.44, the reversal mode changes (Figure 4b). The sense of rotation alternates but the amount of rotation goes from a small angle at one end, to 90° in the middle, to well over 90° at the other end. As the reversal continues on the other side of the transition state, the slower end catches up with the faster end. For reasons that shall become apparent, I will call this the two-domain fanning mode.

[54] Finally, for q ≥ 1.44, reversal occurs in series, with one moment at the end flipping and then its neighbors following it (Figure 5). The minimum energy path between each pair of stable states involves one moment reversing while the others make a small excursion.

Figure 5.

Minimum energy path for three crystals and L = 1.6 W. Same conventions as in Figure 4.

4.2. Four Crystals

[55] If the chain has four crystals, there are five minimum energy paths (Figure 6). For 1 ≤ q ≤ 1.16, there are two stable states and the transition state is symmetric fanning, as in Figure 4a. For 1.18 ≤ q ≤ 1.42, there are three stable states with the middle state having two moments up and two down and a total moment of zero in both directions. For q between 1.18 and 1.22 there is an unusual deviation from perfect alignment with the chain (inset in Figure 6), while between 1.24 and 1.42 the alignment is restored. For all higher aspect ratios the zero-moment states are perfectly aligned with the chain axis. The transition mode going from saturation to either zero-moment state is a two-moment fanning mode in which two moments flip in alternate directions.

Figure 6.

Half of minimum energy paths for four crystals and various aspect ratios. The energy is relative to the saturated state. Only the left half of the path is shown, from negative saturation to zero moment. When the aspect ratio is 1.46, the first transition state almost coincides with the stable state. Inset shows the transition to the zero-moment state for L/W = 1.2.

[56] For aspect ratios greater than 1.44, the set of stable states is the sequence of states that starts with negative saturation and continues by flipping one moment at a time until positive saturation is reached (a domino sequence). However, their connectivity changes. For 1.44 ≥ q ≥ 1.46, the states with one reversed moment are connected only to the saturation states. The lowest-energy path between saturation states skips over these states and jumps directly to the zero-moment state by way of the two-moment fanning mode. For q > 1.48, one moment flips at a time.

4.3. Five Crystals

[57] So far the transition state for the smallest aspect ratios has been symmetric fanning. In five crystals this state disappears, replaced by the two-domain fanning state, as in Figure 4b. This state applies to 1 ≤ q ≤ 1.18. For 1.2 ≤ q ≤ 1.44, there are four stable states, with the middle two having two adjacent moments reversed (normalized moments of ±1). The moments in these states deviate slightly from the vertical for 1.2 ≤ q ≤ 1.24. The reversal state is the two-moment fanning state, as in Figure 6.

[58] The subsequent changes in network are similar to those for four crystals. The domino sequence of stable states appears, but when 1.46 ≤ q ≤ 1.48 the minimum energy path skips over the states with one moment flipped (normalized moments of ±3). When q ≥ 1.5 the sequence is connected by one-moment flips (as in Figure 5).

4.4. Six Crystals and More

[59] The network of stable states for six crystals follows a similar progression to four and five crystals as the aspect ratio changes. The transition states are now as follows: for 1 ≤ q ≤ 1.04, the two-domain fanning state. For 1.06 ≤ q ≤ 1.18 there are three stable states, the middle of which has three moments up and three down, with an off-axis component that is greatest toward the middle of the chain. The transition state is a three-moment fanning state. For 1.2 ≤ q ≤ 1.44, the sequence of flips is two, one, one and two moments. For 1.46 ≤ q ≤ 1.48, the domino sequence appears but initially the states with one moment flipped are skipped. Finally, for q ≥ 1.5, there is a sequence of one-moment flips.

[60] The computational effort rises rapidly with the number of crystals. I only calculated the seven-crystal critical size for an aspect ratio of 1 (see section 4.6). Calculations for eight crystals exceeded the 3 GB memory capacity on the computer that I used.

4.5. Critical Sizes

[61] The critical sizes are plotted in Figure 7 as a function of aspect ratio. The top curve is the critical size for a single crystal with cubic anisotropy and elongation along the crystallographic 〈111〉 axis. It is based on the relaxation calculations by Newell [2006b]. The subsequent curves are for increasing numbers of crystals.

Figure 7.

The critical size LSP, the cube root of the critical volume, as a function of length-to-width ratio. Each curve is for a number of crystals indicated beside the curve. Inset shows the dependence of LSP on the gap (as a fraction of crystal length) for a chain of six crystals and an aspect ratio of 1.

[62] Following the convention established by Butler and Banerjee [1975], the inverse aspect ratio is on the horizontal axis. However, instead of using the Butler and Banerjee [1975] definition of the size of a crystal (its longest side), I define it as the cube root of the volume. This separates effects of volume and shape. In addition to advantages mentioned by Newell and Merrill [1998], this definition allows a simple interpretation of the critical sizes in Figure 7. As the number of crystals in the chain increases, the critical size approaches a lower bound of 10 nm, almost independent of aspect ratio. Crystal interactions have the greatest effect on the critical size for cubic crystals, lowering it by more than 30 nm.

[63] The inset in Figure 7 shows the dependence of the critical size on the spacing between crystals. For small gaps the dependence is linear and weak compared to the difference between the six-crystal and single-crystal sizes. This is a consequence of the weakness of the magnetocrystalline anisotropy in magnetite. It takes a separation of several crystal lengths before the critical size approaches the single-crystal value.

[64] Small jumps in the critical size are evident in Figure 7 at width-to-length ratios of 0.7 or greater. Such jumps are even greater if LSP is calculated using the minimum-energy path (Figure 8a). The minimum-energy paths on each side of the big jump are shown in Figure 8b. Between the two aspect ratios a bifurcation occurs in which two pairs of stable states and transition states appear. This results in a sudden change in the equilibrium distribution of states; naturally, the relaxation rates must also change. Probability can pile up in the intermediate stable states, making it easier to get from negative to positive saturation.

Figure 8.

(a) The critical sizes for a chain of five cubes as a function of the gap between cubes. The gap is expressed as a fraction of cube length. (b) The minimum energy paths for gap ratios on the two sides of the jump in Figure 8a.

[65] Is this jump real? When the new stable state appears, it has a very shallow well around the energy minima. The high-energy barrier approximation, which assumes that local equilibrium is established within each well, is not satisfied. Thus, a jump in LSP indicates a failure of the high-energy barrier approximation. However, an abrupt change in slope can occur at a bifurcation [e.g., Newell and Merrill, 1998].

[66] Fortunately, the jumps are much smaller when the entire network is used to calculate LSP because the probability is distributed among other states that are not bifurcating at the same aspect ratios. Although the minimum-energy network dominates the relaxation rates, other paths become increasingly important as the energies of the equilibrium states converge (as in Figure 3a). In the limit of weak interactions between crystals, all combinations of up and down moments are stable and they all have similar energies. All the relaxation rates converge on the rate for an isolated crystal. In this limit the approximation that calculates Lsp using only the lowest-energy path is least accurate because the higher-energy paths cannot be ignored. However, the divergence between the full-network and minimum-energy calculations is abrupt and occurs as soon as more than two stable states exist.

[67] At large aspect ratios, the critical sizes for two or more crystals converge on each other but do not quite converge on the single-crystal size. That is because in this article I use a fixed value of 109 Hz for the prefactor ν0 (section 3.3) while Newell [2006b] calculates the dependence of the prefactor on the shape of the energy surface and damping. This prefactor varies between about 108 Hz and 1010 Hz, with the larger value occurring at large aspect ratios.

4.6. Strong Interactions: Two-Domain Fanning State

[68] The transition state for an aspect ratio of 1 changes abruptly between four and five crystals, but after that it appears to be converging on a state that is divided into two domains. This state agrees well with the propagating mode of Hendriksen et al. [1994], and the propagation is similar to the movement of a domain wall. However, the details of the propagation only apply to the time evolution of the moments in the limit of high damping, starting from the transition state and going downhill (see also section 3.2.2). I will therefore concentrate on the transition state.

[69] In Figure 9a, the upper half of the moments almost reverse between negative saturation and the transition state. Between the transition state and positive saturation, the lower half catches up and all the moments end up reversed. A rock magnetist looking at the axial component of the moments (Figure 9b) will be irresistibly reminded of a Bloch wall between two ferromagnetic domains. The classic equation for a Bloch wall [Landau and Lifshitz, 1935] involves a tanh function, and such a function is a good fit to the z components. Otherwise, the structure is nothing like a domain wall: the x components oscillate between each crystal and the next.

Figure 9.

(a) The two-domain fanning state for seven crystals. (b) Axial component of the moments two-domain fanning state for five to seven crystals. The vertical axis is the distance from the center of the chain, divided by the width of a crystal. The horizontal axis is the z component of the moment of each crystal. These profiles are converging on a state that is well approximated by the curve mz = tanh(1.4 z) (red curve). (c) SP critical size for two-domain fanning state as a function of the number of crystals in a chain (odd numbers only).

[70] The tanh function provides a good initial guess for the two-domain fanning state in larger chains. I refine this guess using a Levenberg-Marquardt algorithm to solve the equilibrium equations (implemented in MATLAB using fsolve). For odd numbers of crystals, the two-domain fanning mode remains a transition state up to chain lengths of 79 crystals (the largest number that I checked). If there is an even number of crystals, it is a transition state up to 10 crystals and then it becomes stable. In that case the network connections cannot be extrapolated from smaller numbers of crystals.

[71] Restricting the analysis to odd numbers of crystals, we see that the SP critical size approaches a lower limit of about 10.2 nm (Figure 9c).

4.7. Comparison With Data

[72] For a comparison with measured sizes, it is easiest to return to the conventional definition of size as the longest dimension of the crystal. Most summaries of the sizes are of the longest size and aspect ratio, and these cannot be converted to a distribution of the cube root of the volume unless the individual measurements are known. For a chain of six crystals, gap fractions η between 0 and 1, and Q = W/L between 0.1 and 1, an approximate formula for the critical length is

equation image

The error in this approximation is at most 0.4 nm, and is generally much less.

[73] In TEM images of magnetosome crystals, the crystals in a chain often appear to be touching. This is probably an artifact of the viewing angle (M. Winklhofer, personal communication, 2009). There must be room for the magnetosome membranes, which Gorby et al. [1988] found were 3.7 nm thick in Magnetospirillum magneticum strain MS-1. Their estimate of the spacing was 9 nm. The corresponding gap fraction for a cube at the critical size is 0.6.

[74] The SP critical size for six crystals and gap fractions of 0 and 0.6 are plotted in Figure 10. The single-crystal upper and lower sizes, based on calculations by Newell [2006b] and Newell and Merrill [1999] for spheroids, are included for comparison. The size for these spheroids is defined as the length of a cuboid with the same volume. It might seem more realistic to use numerical estimates of the upper critical size for cuboids [Fabian et al., 1996; Witt et al., 2005; Muxworthy and Williams, 2006] and elongated cubo-octahedra [Witt et al., 2005], but these estimates are hard to reconcile. The estimates for cuboids differ significantly, but all of them are much smaller than the estimate for the elongated cubo-octahedra. Indeed, the latter is comparable to the upper critical size for the chain of cuboidal crystals. A possible reason for this is that the extra faces truncate the sharp corners of the cuboids, reducing the magnetic fields. In response to the fields in the corners, the magnetization adopts a nonuniform “flower” structure that may ease further deviations from the single-domain state. If this argument is correct, the spheroids should have the largest critical size because they allow perfectly uniform magnetization. Instead, the upper critical size of the cubo-octahedra is largest.

Figure 10.

Critical crystal lengths for a chain of six crystals and intercrystal gaps equal to 0 (η = 0) and 0.6 L (η = 0.6). Also shown are critical sizes for isolated crystals: a SP size based on Newell [2006b] and a MD size from Newell and Merrill [2000b]. The upper critical size is for a chain of three crystals [Muxworthy and Williams, 2006]. The light blue dots are measured lengths and aspect ratios of magnetosome crystals [Petersen et al., 1986; Vali et al., 1987; Mann et al., 1987; Hesse, 1994; McNeill, 1990; Petersen et al., 1992; Peck and King, 1996; Iida and Akai, 1996; Lean and McCave, 1998; Isambert, 2005]. The purple rectangles are approximate two-sigma bounds for crystals from M. magneticum [Devouard et al., 1998; Kopp et al., 2006] while the green rectangles are the same for crystals from other strains of magnetotactic bacteria [Meldrum et al., 1993a, 1993b; Thomas-Keprta et al., 2000].

[75] Numerical estimates of the upper critical size are obtained as follows. For each particle size an initial guess is provided for the micromagnetic solver. This guess may be either the magnetization state from a smaller crystal or a single-domain state. If the solver fails to converge on a single-domain state, it is assumed that the single-domain state is unstable. The results may depend as much on the numerical procedure as on the physics. By contrast, the upper critical size for spheroids [Newell and Merrill, 1999] is based on a rigorous perturbation analysis [Aharoni, 1959, 1966, 1997]. I will therefore base my interpretation on Newell and Merrill [1999]. Despite these reservations, I include a numerical calculation by Muxworthy and Williams [2006] for a chain of three cuboids.

[76] Included in Figure 10 are boxes representing measured sizes and aspect ratios of various strains of magnetotactic bacteria: magnetospirillia [Devouard et al., 1998; Kopp et al., 2006], vibrios [Meldrum et al., 1993b; Devouard et al., 1998; Thomas-Keprta et al., 2000], and cocci [Meldrum et al., 1993a]. I use two-sigma bounds, if provided by the authors, or estimate these bounds if only histograms of sizes are available.

[77] Also included in Figure 10 are about 3000 individual measurements of aspect ratios and lengths. These are mostly magnetofossils from marine and lacustrine sediments. They are identified by their distinctive shapes and presence in chains. One data set for living bacteria [Mann et al., 1987] gives sizes and shapes of crystals that are still growing. The growing crystals are cubo-octahedral up to about 20 nm, after which they grow faster in the [11equation image] direction, an unusual direction for bacterial magnetite. Many more points cluster on the L = W axis, most of which are from Lean and McCave [1998]. They form a distribution that seems distinct from the large cluster at modest aspect ratios. It is not clear whether this is a measurement artifact or a particular strain of bacteria.

[78] The magnetofossils tend to be larger than the magnetite from cultured bacteria. This may reflect the difference between the natural environment and the conditions in which the bacteria are cultured. Few of the magnetofossils are in the single-crystal SP size range, and about half would be MD without the interactions with other crystals in the chains.

5. Discussion

[79] Most of the crystal sizes and shapes in Figure 10 are above the critical SP size for an isolated crystal. The main exceptions are the Magnetospirillum species. The two purple boxes in Figure 10 represent data sets for Magnetospirillum magneticum strains MS-1 [Devouard et al., 1998] and AMB-1 [Kopp et al., 2006]. If strain AMB-1 starts with no magnetic material and grows several crystals simultaneously, it establishes a similar range of sizes and shapes within 28 h [Li et al., 2008]. The other Magnetospirillum species, M. gryphiswaldense, has an apparently bimodal distribution of shapes and sizes [Moisescu et al., 2008]. The isometric crystals are all small enough to be SP in isolation while another fraction with length-to-width ratios between about 0.7 and 0.95 is mostly above the critical size.

[80] In electron micrographs, the variety of apparent shapes is consistent with different views of randomly oriented crystals. The random orientations may be due to the preparation of the samples. Measurements of length and width are based on two-dimensional silhouettes of the crystals and can be misleading. None of the size estimates for these crystals come with information on the orientation, and in any case there is no unique definition of size for such shapes. In simulations of randomly oriented cubo-octahedral crystals I found that they can have apparent length-to-width ratios less than 0.7 and lengths that are 50% greater than the cube root of the volume. Thus, the crystals represented by the two purple boxes in Figure 10 could be isometric cubo-octahedra with dimensions smaller than the SP/SD critical size for an isolated crystal.

[81] All Magnetospirillum species live in fresh water, prefer microaerobic conditions and have cubo-octahedral magnetite crystals [Spring and Bazylinski, 2006]. They also navigate using axial magnetoaerotaxis, which determines which way to swim based on the time dependence of the oxygen concentration [Frankel et al., 1997]. The “large majority” of magnetotactic bacteria use polar magnetoaerotaxis [Frankel et al., 2007], choosing the swimming direction based only on the current oxygen concentration.

[82] The rest of the data in Figure 10 are taken from marine sediments or cultured bacteria that come from marine environments. Of the cultured strains, vibroid strain MV-1 [Meldrum et al., 1993b; Devouard et al., 1998] has crystals with lengths down to 15 nm, below the SP/SD critical size for an isolated crystal. Another strain, MV-4, has a similar cubo-octahedral morphology to the magnetospirillia but substantially larger sizes. The rest of the cultured strains have crystals that are large enough to be SD or MD in isolation. Most of the crystals from cultured bacteria are smaller than the magnetofossils represented by the blue dots in Figure 10. This may be because the growing conditions for the cultured bacteria are not optimal, or it may be because they are different species. However, there is a population of small cubo-octahedral magnetofossils that are small enough to be SP in isolation.

[83] Some studies of magnetofossils find a dependence of crystal shape on the chemical environment [Hesse, 1994; Yamazaki and Kawahata, 1998; Lean and McCave, 1998], particularly the organic carbon flux and oxygen supply. Nearly isometric particles are more common in high-oxygen environments and can account for anywhere from 0% to 50% of the magnetofossils. These changes in the physical properties may be linked to ecological factors such as organic carbon content of sediments [Hesse, 1994; Lean and McCave, 1998; Kopp and Kirschvink, 2008], and affect the magnetic properties of the magnetofossils [Hanzlik et al., 2002]. Such a separation into two distinct populations also seems to occur in lakes, based on a statistical analysis of magnetic remanence curves [Egli, 2004].

[84] About half of the magnetofossils are large enough to be MD unless they are in chains. They are therefore significantly larger than they need to be to carry a stable magnetic moment. Does this imply that evolution failed to optimize the machinery for magnetotaxis in these organisms? It does not. The amount of iron in the bacterium is proportional to the total magnetic moment, not the size of an individual crystal.

[85] A possible advantage of growing larger crystals is that they will lower the SP/SD critical size for new crystals growing at the ends of a chain. However, this advantage is small. A 10 μm cube has only 1/125 the volume, and therefore the moment, of a 50 μm cube, so its contribution to the total moment of the chain is insignificant.

6. Conclusions

[86] The new method of calculating energy barriers using homotopy continuation is very powerful and robust. Its main limitation is the sheer number of stationary states in the energy. In the limit of N weakly interacting crystals, there are 2N stable states and 2N transition states connecting them. However, as the strength of the magnetostatic interaction increases, the energies of these states diverge and most of the stable states become unstable. If this method can be adapted so it ignores all but the relevant states, it will be very powerful.

[87] If SP crystals are isolated, their moments flip from up to down and back independently; but as the magnetostatic coupling increases, the paths between positive and negative saturation become fewer and more coherent. For isometric crystals there is a coherent reversal mode. In chains of four crystals or less the transition state is a symmetric fanning state in which the single-crystal moments are rotated alternately clockwise and anticlockwise by 90°. In chains of five or more crystals there is a new mode in which fanning propagates from one end of the chain to the other. I call the transition state the “two-domain fanning state.”

[88] The superparamagnetic critical size rapidly approaches a limit as the number of crystals increases. For crystals that are in contact, the cube root of the critical volume is about 10 nm, almost independent of aspect ratio. This critical size has a remarkably weak dependence on the distance between crystals, even in the limit of no elongation. Typical lengths of magnetosome chains being over 10, there should be little effect on the stability when a chain is divided among offspring. The main advantage of the chains should be to stabilize the moment in new magnetosome crystals as they are growing.

[89] Most of the magnetosome crystals are well above this size limit. About half are even large enough to be multidomain if the magnetization is not kept uniform by the magnetostatic coupling in a chain. However, in both marine and freshwater environments there is also a population of bacteria manufacturing cubo-octahedral crystals, and a large fraction of these crystals are small enough to be superparamagnetic if they are not stabilized by magnetostatic coupling. In fresh water the bacteria producing these crystals are from the genus Magnetospirillum. These bacteria may also be the only bacteria that use axial magnetoaerotaxis rather than polar magnetoaerotaxis. In marine environments the presence of cubo-octahedral crystals is related to the redox conditions, and their presence in fresh water may also be related to environmental conditions.


[90] This research was funded by the NASA Exobiology Division and the National Science Foundation Geophysics Program. I would like to thank Michael Winklhofer and Ramon Egli for comments that led to substantial improvements in the manuscript; I would also like to thank Ramon Egli for the database of crystal sizes used in Figure 10. Thoughtful reviews by Mike Jackson and Karl Fabian also led to substantial improvements.