Static toy model

Fig. 2 shows a very simple model of cell in compression, consisting of two rigid, slender rods (length l_i,  radius a_i,  aspect ratio ϵ ≡ a_i/l_i) connected by a torsional spring (hook) of constant modulus κ and initially aligned along the x-axis. An external force F_P of constant magnitude F_P pushes the flagellum along the unit vector p toward the body—this models the reaction force from the fluid to the propulsive force (thus the subscript P) exerted by the flagellum. The “front” end of the body is pinned at the origin while free to rotate. This situation models a cell whose body is stuck to a surface. We will consider this model in two special cases: 1) p = −e_x,  with the flagellum end constrained to lie on the x-axis, analogous to compression of a beam with a horizontal force load on both ends, and 2) p = −p_f,  a closer representation of a swimmer generating propulsion aligned with the flagellar axis. Without loss of generality, motion is restricted to the x − y plane, and torques are in the n = e_z direction. The segment orientations are parametrized by angles θ_i,  i.e., p_i = [cosθ_i sinθ_i]^T,  and we define the bending angle θ = θ_f − θ_b. From Eq. 1, the translational velocity of component i is:

where X is a constraint force exerted by the hook to maintain connectivity, X_C is the external constraint fixing the body end position, and δ is the Kronecker delta. For slender rods, the translational resistance is A_i = ζ^∥p_ip_i + ζ^⊥(δ − p_ip_i) where δ is the identity tensor, η is the fluid viscosity, and ζ^⊥ = 4πη(l_i/2)/ln(ϵ⁻¹) with 2ζ^∥ = ζ^⊥. The angular velocities are given by Eq. 2:

where Ci⊥=(2/3)πηli3/ln(ϵ−1) is the rotational resistance normal to the plane. The first torque contribution is the linear contribution from the spring and the second includes all force moments. Two algebraic constraints arise: the first maintaining connectivity (Eq. 3) and the second fixing the following body-end position:

Case 1 requires an additional constraint force X^′ = X^′e_y fixing the flagellum end to the x-axis so that

Equations 3, 5, 6, 7, and 8 are nondimensionalized with characteristic length l_b/2,  force F_p,  torque F_Pl_b,  and time ζ^∥(l_b/2)/F_P. We define the length ratio between flagellum and cell body as L = l_f/l_b. The following dimensionless differential-algebraic equation summarizes the dynamics of the static toy model (^∗ denotes dimensionless variables):

Here a dimensionless group Fl_F = F_Pl_b/2κ arises, which we call the flexibility number. The subscript F denotes a definition based on force. (Below we define one based on motor torque.) It characterizes the ratio of the force moment to the bending resistance. We wish to examine the effects of Fl_F on the existence and stability of static equilibrium points arising from Eq. 9. Our choice of p affects the equilibria we find. In either case, a trivial solution to Eq. 9 maintaining static equilibrium is r_b = e_x,  r_f = (2 + L)e_x,  θ_b = θ_f = 0,  X = −X^C = −e_x,  X^′ = 0. Linearizing Eq. 9 around this point and seeking solutions of the form exp(μt),  we find that μ satisfies the following:

_b,  θ_f), denoting a buckled conformation. This buckling eigenvalue changes sign when Fl_F exceeds a critical value as follows:

indicates a pitchfork bifurcation. We note that although the loading chosen for Case 1 is chosen to mirror a classical buckling problem, it is not a representative picture of how a swimmer functions.

(a) Static toy model showing the body and flagellum as two connected rods. Propulsion is either aligned with the x-axis (Case 1) or the flagellum (Case 2). (b) Dynamic toy model shows a spheroidal body and cylindrical flagellum. The propulsion is aligned with the latter. The cylinder is exaggerated for clarity, and the associated helix is the same tapered one as Fig. 1. To see this figure in color, go online.

For the more realistic Case 2, there is always a zero eigenvalue that arises from rotational invariance—every angle the swimmer makes with the anchor point is equivalent. The remaining eigenvalue corresponds to the buckled state but is always negative, unlike Case 1. Thus a swimmer pinned by its “nose” does not buckle. To understand this result, we note that static equilibrium requires X = −p_f,  and because the propulsive force is parallel to p_f,  this condition can only hold for θ = 0. Any nonzero value of θ will result in a restoring spring torque that will drive the system to the state θ = 0. In other words, because the propulsive force is parallel to p_f,  it can generate no torque to counteract the restoring torque so any initially bent conformation will straighten over time. With the next toy model we address how this result changes when the cell can swim freely.