Abstracts of Publications:
A catalogue of stable equilibria of planar extensible or inextensible elastic rods for all possible Dirichlet boundary conditions R.S. Manning (preprint, submitted to Journal of Elasticity)
We catalogue configurations that locally minimize energy for a planar elastic rod (extensible-shearable
or inextensible-unshearable) subject to arbitrary Dirichlet boundary conditions in position and orientation.
Via a combination of
analysis and computation, we determine several bifurcation surfaces in the 3-parameter space
of boundary conditions and explore how they depend on the rod material parameters, including in
the inextensible limit. For each possible boundary condition, we find all stable equilibria with
sufficiently low energy that they might be competitive within a Boltzmann distribution
if the rod were used to model DNA with tens or hundreds of base pairs, the length-scale
relevant for DNA looping.
Depending on the boundary conditions, there are as many as three such equilibria.
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Thermodynamic Analysis of Self-Assembly in
Coiled-Coil Biomaterials B.P. Tsang, H.S. Bretscher, B. Kokona, R.S. Manning and R. Fairman, Biochemistry 50 (2011) 8548-8558.
Coiled-coil protein structural motifs have
proven amenable to the design of structurally well-defined
biomaterials. Mesoscale structural properties can be fairly well
predicted based on rules governing the chemical interactions
between the helices that define this structural motif. We
explore the role of the hydrophobic core residues on the selfassembly
of a coiled-coil polymer through a mutational
analysis coupled with a salting-out procedure. Because the
resultant polymers remain in solution, a thermodynamic
approach is applied to characterize the polymer assembly
using conventional equations from polymer theory to extract
nucleation and elongation parameters. The stabilities and
lengths of the polymers are measured using circular dichroism spectropolarimetry, sizing methods including dynamic light
scattering and analytical ultracentrifugation, and atomic force microscopy to assess mesoscale morphology. Upon mutating
isoleucines at two core positions to serines, we find that polymer stability is decreased while the degree of polymerization
is about the same. Differences in results from circular dichroism and dynamic light scattering experiments suggest the presence
of a stable intermediate state, and a scheme is proposed for how this intermediate might relate to the monomer and polymer
states.
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Ineffective perturbations of a
planar elastica K.M. Peterson and R.S. Manning, Involve 2 (2009) 557-578.
The bifurcation diagram for the buckling of a planar elastica under a load lambda
is made up of a "trivial" branch of unbuckled configurations for all lambda and a sequence
of branches of buckled configurations that are connected to the trivial branch at pitchfork bifurcation
points. We use several perturbation expansions to determine how this diagram perturbs with
the addition of a small intrinsic shape in the elastica, focusing in particular on the
effect near the bifurcation points.
We find that for almost all intrinsic shapes epsilon f(s), the difference between the buckled
solution and the trivial solution is O(epsilon^(1/3)), but for some ``ineffective'' f,
this difference is O(epsilon), and we find functions u_j(s) so that f is ineffective at
bifurcation point number j when < f, u_j > = 0.
These ineffective perturbations have important consequences in numerical simulations,
in that the perturbed bifurcation diagram has sharper corners near the former
bifurcation points, and there is a higher risk of a numerical simulation
inadvertently hopping between branches near these corners.
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An extended conjugate
point theory with application to the stability of planar buckling of an elastic
rod subject to a repulsive self-potential K.A. Hoffman and R.S. Manning,
SIAM Journal on Mathematical Analysis 41 (2009) 465-494.
The theory of conjugate points in the classic calculus of variations
allows, for a certain class of functionals,
the characterization of a critical point as stable (i.e., a local minimum) or not.
In this work, we generalize this theory to more general functionals,
assuming certain generic properties of the second variation operator.
The extended conjugate point theory is then applied to a
two-dimensional elastic rod subject to pointwise self-repulsion.
The critical points are computed by
numerically solving first-order integro-differential equations using a
finite difference scheme. The stability of each
critical point is then computed by determining conjugate points of the
second variation operator. In addition, the generalized theory requires
the numerical evaluation of the crossing velocity of the zero eigenvalue
of the second variation operator at each conjugate point, a feature not
present in the classic case (where the crossing velocity can be shown to
always be negative). Results demonstrate that the repulsive potential
has a stabilizing influence on some branches of critical points.
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Conjugate Points Revisited and Neumann-Neumann Problems
R.S. Manning, SIAM Review 51 (2009) 193-212.
The theory of conjugate points in the calculus of variations is reconsidered
with a perspective emphasizing the connection to finite-dimensional optimization.
The object of central importance is the spectrum of the second-variation
operator, analogous to the eigenvalues of the Hessian matrix in finite dimensions.
With a few basic properties of this spectrum, one can see the logic behind
the classic result that "stability requires the lack of conjugate points".
Furthermore, we show how the spectral perspective allows the easy extension of
the conjugate point approach to variants of the classic problems in the literature,
such as problems with Neumann-Neumann boundary conditions.
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Quantitative atomic force microscopy image analysis of unusual
filaments formed by Acanthamoeba castellanii myosin II rod domain
D.J. Rigotti, B. Kokona, T. Horne, E.K. Acton, C.D. Lederman, K.A. Johnson, R.S. Manning,
S.A. Kane, W.F. Smith, and R. Fairman, Anal. Biochem.
346 (2005) 189.
We describe a quantitative anlysis of
Acanthamoeba castellanii myosin II rod domain images collected
from atomic force microscope experiments. These images reveal that the rod domain
forms a novel filament structure, most likely requiring unusual
head-to-tail interactions. Similar filaments are aseen also in negatively
stained electron microscopy images. Truncated myosins from Acathamoeba and other
model organisms have been visualized before, revealing laterally associated
bipolar minifilaments. In contrast, the filament structures that we observe are dominated
by axial rather than lateral polymerization. The unusually small features in this
structure (1-5 nm) required the development of quantitative and statistical techniques for
filament image analysis.
These techniques enhance the extraction of features that hitherto have been difficult
to ascertain from more qualitative imaging approaches.
The heights of the filaments are observed to have a bimodal distribution consistent with the
diameters of a single rod domain and a pair of close-packed rod domains.
Further quantitative analysis indicates that in-plane association is limited to at most
a pair of rod domains. Taken together, this implies that the filaments contain no more
than four rod domains laterally associated with one another, somewhat less than seen in
bipolar minifilaments. Analysis of images of the filaments decorated with an anti-FLAG
antibody reveals head-to-tail association with mean distances between the antibodies
of 75 +/- 15 nm. We consider a set of molecular models to help interpret possible
structures of the filaments.
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Link, Twist, Energy, and the Stability of DNA Minicircles
K.A. Hoffman, R.S. Manning, and J.H. Maddocks, Biopolymers ,
70 (2003) 145.
We describe how the stability properties of DNA minicircles can be
directly read from plots of various biologically intuitive quantities
along families of equilibrium configurations. Our conclusions follow
from extensions of the mathematical theory of distinguished
bifurcation diagrams that are applied within the
specific context of an elastic rod model of minicircles. Families of
equilibria arise as a twisting angle alpha is varied. This angle
is intimately related to the continuously varying linking number Lk for nicked
DNA configurations that is defined as the sum of Twist and Writhe. We
present several examples of such distinguished bifurcation diagrams
involving plots of the energy E, linking number Lk, and a twist
moment m3, along families of cyclized equilibria of both
intrinsically straight, and intrinsically curved, DNA fragments.
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Stability of an elastic rod buckling into a soft wall
R.S. Manning and G.B. Bulman, Proc. R. Soc. Lon. Ser. A, 461 (2005) 2423.
The conjugate point theory of the calculus of variations is extended to apply to
the buckling of an elastic rod in an external field,
using the operator approach presented in (Manning, Rogers, and Maddocks, Proc. Roy. Soc. A 454 (1998) 3047),
which we show can be used when the second variation operator is an integrodifferential
operator, rather than a differential operator as in the classical case.
The external field is chosen to model two parallel "soft" walls.
We consider the examples of 2D buckling under both pinned-pinned
and clamped-clamped boundary conditions, as well as the 3D clamped-clamped
problem, where we consider the importance of the rod cross-section
shape as it ranges from circular to extreme elliptical.
For each of these problems, we find that in the appropriate limit,
the soft-wall solutions approach a "hard-wall" limit, and so
we make conjectures about these hard-wall contact equilibria and their stability.
In the 2D pinned-pinned case, this allows us to assign stability
to the configurations reported in (Holmes, Domokos, Schmitt, Szebernyi, Comput. Methods Appl. Mech. Engrg. 170
(1999) 175) and reconsider the experimental results discussed therein.
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Calculation of the stability index in parameter-dependent
calculus of variations problems: Buckling of a twisted elastic strut
K.A. Hoffman, R.S. Manning, and R.C. Paffenroth,
SIAM Journal on Applied Dynamical Systems 1 (2002) 115.
We consider the problem of minimizing the energy of an inextensible elastic strut
with length 1 subject to an imposed twist angle and force.
In a standard calculus of variations approach, one first locates equilibria by
solving the Euler-Lagrange ODE with boundary conditions at arclength
values 0 and 1. Then, one classifies
each equilibrium by counting conjugate points, with local minima
corresponding to equilibria with no conjugate points.
These conjugate points are arclength values sigma <= 1 at which a second ODE
(the Jacobi equation) has a solution vanishing at 0 and sigma.
Finding conjugate points normally involves the numerical
solution of a set of initial value problems for the Jacobi equation.
For problems involving a parameter lambda,
such as the force or twist angle in the elastic strut,
this computation must be repeated for every value of lambda
of interest.
Here we present an alternative approach that takes advantage
of the presence of a parameter lambda.
Rather than search for conjugate points sigma <= 1 at a fixed
value of lambda, we search for a set of special parameter values
lambdam (with corresponding Jacobi solution zetam)
for which sigma=1 is a conjugate point.
We show that under appropriate
assumptions, the index of an equilibrium at any lambda
equals the number of these zetam for which
the inner product of zetam
with S zetam is negative, where
S is the Jacobi differential operator at lambda.
This computation is particularly simple when
lambda appears linearly in S.
We apply this approach to the elastic strut, in which the
the force appears linearly in S,
and as a result, we locate the conjugate points for any twisted unbuckled
rod configuration without resorting to numerical solution of differential equations.
In addition, we numerically compute 2-dimensional sheets of buckled equilibria (as the two
parameters of force and twist are varied)
via a coordinated family of one-dimensional parameter continuation computations.
Conjugate points for these buckled equilibria are determined by numerical
solution of the Jacobi ODE.
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Stability of $n$-covered circles for elastic rods with constant planar intrinsic curvature ,
R.S. Manning and K.A. Hoffman, Journal of Elasticity 62 (2001) 1.
A stability index is computed for the n-covered circular equilibria
of inextensible-unshearable elastic rods with constant planar intrinsic
curvature u and constant values for the twisting stiffness and two
bending stiffnesses. A simple expression is derived for the index as
a function of u, p (the ratio of bending stiffness out of the plane of curvature to
bending stiffness in the plane of curvature), and g (the ratio of
twisting stiffness to bending stiffness in the plane of curvature).
In particular, for intrinsically straight rods
(u=0) we prove that the 1-covered circle is stable if and
only if p >= 1, and the n-covered circle (n>1) is stable
if and only if
g>1, p>1, and (n-1)2/n2 <= (g-1)(p-1)/(gp).
The index is computed by framing the standard Euler-Cosserat equilibrium
equations within a constrained variational principle with an
isoperimetric constraint ensuring the ring closure.
The fact that u appears linearly in the second variation
allows the second
variation to be diagonalized using the eigenvectors of an appropriate
eigenvalue problem similar to a Sturm-Liouville problem. This
diagonalization allows the direct computation of an unconstrained
index (disregarding ring closure). We then apply a result of Maddocks
to find the constrained index in terms of this unconstrained index and a
correction computable from the linearized constraint.
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Multiple Equilibria in DNA rings,
P. Furrer, R.S. Manning and J.H. Maddocks, to appear in
Biophysical Journal
Within the context of DNA rings,
we analyze the relationship between intrinsic shape and
the existence of multiple stable equilibria,
either nicked, or cyclized with the same link.
A simple test, based on a perturbation expansion of
symmetry-breaking within a continuum elastic rod model,
provides good predictions of the occurrence of such multiple
equilibria.
The reliability of these predictions is verified
by direct computation of nicked and cyclized equilibria
for several thousand DNA minicircles
with lengths of 200 and 900 basepairs.
Further,
our computations of equilibria for nicked rings
predict properties of the equilibrium distribution
of link, as calculated by much more
computationally intensive Monte Carlo simulations.
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Symmetry breaking and the Twisted Elastic Ring,
R.S. Manning and J.H. Maddocks,
Computer Methods in Applied Mechanics and Engineering 370 (1999) 313.
Symmetry breaking is considered in the context of a parameter-dependent
two-point boundary value problem describing a twisted elastic ring
that arises as a model of DNA minicircles.
We explicitly determine, via a perturbation expansion, those representatives
of a manifold of solutions to a symmetric BVP that persist
when the symmetry is broken. Conditions generating a generic
splitting are identified, and some degenerate cases are studied.
In both the generic and degenerate cases,
the results of the perturbation expansion are used to
improve the efficiency of existing algorithms for computing
symmetry-broken bifurcation diagrams.
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Isoperimetric conjugate points with application to the stability of DNA
minicircles, R.S. Manning, K.A. Rogers and J.H. Maddocks,
Proceedings of the Royal Society of London, Series A 454 (1998) 3047.
A conjugate point test determining an index of the constrained
second variation in one-dimensional isoperimetric calculus of
variations problems is described. The test is then
implemented numerically to determine stability
properties of equilibria within a continuum mechanics model of DNA
minicircles.
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Interactive computation, parameter continuation, and visualization,
J.H. Maddocks, R.S. Manning, R.C. Paffenroth, K.A. Rogers, and
J.A. Warner, Int. J. Bif. Chaos 7 (1997) 1699.
Nonlinear problems arising in modelling applications are frequently parameter dependent,
so that families of solutions are of interest. Such problems naturally lend
themselves to interactive computation that exploits
parameter continuation methods combined with visualization
techniques. Visualization provides both understanding of the solution set
and feedback for computational steering.
We describe various issues that have arisen in our investigations of
problems of this general type.
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A Continuum Rod Model of Sequence-Dependent DNA Structure,
R.S. Manning, J.H. Maddocks and J.D. Kahn, J. Chem. Phys. 105 (1996) 5626.
Experimentally motivated parameters from a base-pair-level
discrete DNA model are averaged to yield parameters for a
continuum elastic rod with a curved unstressed
shape reflecting the local DNA geometry.
The continuum model permits computations
with discretization lengths longer than the intrinsic discretization of the
base-pair model, and, for this and other reasons, yields an efficient
computational formulation. Obtaining continuum stiffnesses is
straightforward, but obtaining a continuum unstressed shape
is hindered by the "noisy" small-scale structure and rapid helix
twist of the discrete unstressed shape. Filtering of the discrete data and
an analytic transformation from the true normal-vector field to a
natural (untwisted) frame allows a stable continuum fit.
Equilibrium energies of closed rings predicted by the continuum
model are found to match the energies of the underlying discrete
model to within 0.5%.
The model is applied to a set of 11 short
DNA molecules (approximately 150 bp) and properly distinguishes their
cyclization probabilities (J factors) when compared both to experimental
cyclization rates and to Monte Carlo simulations. The continuum
model does not include entropic contributions to the free energy.
However, because of its rapid and accurate computation of internal energy,
the continuum model should, when
combined with further work on entropic effects, be a useful method for
computing experimental DNA free energies.
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A uniform regularized semiclassical propagator for the 1/x^2
potential, R.S. Manning and G.S. Ezra, Phys. Rev. A, 53 (1996) 661.
We apply recent methods for semiclassical time propagation
involving non-Cartesian variables to the repulsive 1D potential
V(x)=x-2, x >= 0. In order to properly treat non-Cartesian
variables, a quantum regularization is first performed which
leads to a Langer-type potential correction term in the
Gutzwiller-Van Vleck propagator. A non-uniform semiclassical
treatment of V(x)=x-2 using this regularization gives
an improvement to earlier unregularized results, and a uniform
regularized propagator is very nearly exact for all times.
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Regularized semiclassical radial propagator for the Coulomb potential,
R.S. Manning and G.S. Ezra, Phys. Rev. A, 50 (1994) 954.
We derive a regularized semiclassical radial propagator for the Coulomb
potential, a case for which standard approaches
run into well-known difficulties associated with a
non-Cartesian radial coordinate and a potential singularity.
Following Kleinert [ Path Integrals in Quantum Mechanics,
Statistical Mechanics and Polymer Physics (World Scientific, Singapore, 1990)],
we first perform a quantum-mechanical
regularization of the propagator.
The semiclassical limit is then obtained by stationary phase.
The semiclassical propagator so derived has the
standard Van Vleck-Gutzwiller form for the radial Coulomb problem
with a potential correction (Langer modification) term included.
The regularized semiclassical propagator is applied to
compute the autocorrelation function for a
Gaussian Rydberg wavepacket.
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Theory of Projected Probabilities on Non-Orthogonal States:
Application to Electron Populations in Molecules,
R.S. Manning and N. De Leon, J. Math. Chem., 5 (1990) 323.
Often it is important to consider the expansion of a quantum state z in terms
of physically meaningful basis states. For example, molecular orbitals can
be expressed as linear combinations of atomic orbitals, or vibrational states
can be expressed as superpositions of local or normal mode eigenstates.
In such expansions, it then becomes desirable to determine how much
"character" a quantum state has in one of these basis states. One way of
accomplishing this task is to calculate the projected probability of
z on basis state j. In this paper, we consider this general quantum
mechanical problem. If the basis states are orthonormal, then the projected
probability of z on j is of course the square of their inner product.
However, if the basis states are not orthogonal, then this result is no longer
valid and one must develop a more general theory to calculate these
projected probabilities. An earlier paper used one-dimensional projection
operators to initiate this theory and gave closed-form results
for the case of two non-orthogonal basis states (DeLeon and Neshyba,
Chem. Phys. Lett. 151 (1988) 296). One- and many-dimensional projection
operators, together with linear algebraic techniques, are used to extend
this theory to the n non-orthogonal basis state case. Explicit closed-form
results are given for the two- and three-state cases, and a general
algorithm is developed for the case of four or more basis states. Application
of the theory is made to atomic populations in three- to six-atom
molecules, and comparisons are made to the related work of Mulliken.
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Filtration of sickle cells: recruitment into a rigid fraction
as a function of density and oxygen tension,
E.A. Schmalzer, R.S. Manning, and S. Chien, J. Lab. Clin. Med., 113 (1989) 727.
The rheological behavior of sickle cells was evaluated with a constant
flow filtration system at 37o C by using filters with
pores of 4.8 micrometer diameter. Analysis of the shape of the
pressure-time curves suggested that sickle cells were subdivided
into two discrete subpopulations: (1) relatively deformable cells
and (2) cells so rigid that they plugged the pores. The analysis
allowed calculation of the fraction of cells causing plugging F, even
when F represented less than 0.01% of the population. F rose with
falling O2 pressure level, apparently with the recruitment
of progressively less dense cells into the plugging fraction. A density
profile for each sample was constructed. When we made the assumption
that the plugging cells represented the denser moiety, this profile
allowed identification of the threshold O2 pressure level at
which cells of a given density, deoxygenated under these conditions, became
unable to traverse the 4.8 micrometer pore.
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rmanning@haverford.edu