To compute the folded structure of a protein using a physical
model and the monomer sequence has been regarded as a challenge because it involves
multiple time and space scales. Our approach has been to understand how proteins
physically fold and to try to apply that strategy to protein structure prediction.
We believe that proteins break their large global optimization problem into
smaller local optimizations. We have been exploring various methods for exploiting
that idea for protein structure prediction.
Prototype nanoelectromechanical devices incorporating individual
multiwall carbon nanotubes as torsion bars or rotary bearings have been fabricated
and tested by various groups. Typical length scales are: 1 micron for the overall
span, 25 nm for the diameter, 0.5 nm for the interwall gap. The experimental
evidence collected so far is puzzling, pointing out a need for a better understanding
of the interwall mechanical coupling mechanisms. We speculate that the basic
mechanism behind progressive interwall coupling is the formation of bridging
defects, that is, covalent links between adjacent walls, triggered by inward
migration of chromium atoms (which are evaporated onto the outer wall of the
nanotube when fabricating the device).
The focus of this talk is to develop continuum mechanics type
of models for studying the deformation of nano-scale objects such as nano-tubes,
nano-rods and DNA. The basic tool we will use is various generalizations of
the classical Cauchy-Born rule. For this purpose, we will first review the classical
Cauchy-Born rule for bulk crystals. We will discuss the validity of the Cauchy-Born
rule and give a precise characterization of its boundary of invalidity. We then
discuss the generalization of the Cauchy-Born rule to curved low dimensional
objects, including the local Cauchy-Born rule and the exponential Cauchy-Born
rule. Finally we turn our attention to nano-tubes and nano-rods, and examine
whether their deformation can be described by these continuum theories.
Activated trajectories are paths that pass over a significant energy barrier
between two (or more) stable states. A typical activated trajectory "incubates"
in one of the wells for a significant time period, but once the process is initiated
the transition time can be extremely short. A boundary value formulation is
proposed that computes efficiently rare and activated trajectories. A large
integration step is used in two examples: the Mueller potential, and a conformational
transition in alanine dipeptide, a model for protein backbone.
An algorithm is presented to compute time scales of complex processes
following predetermined milestones along a reaction coordinate. A non-Markovian
hopping mechanism is assumed and constructed from underlying microscopic dynamics.
General analytical analysis, a pedagogical example, and numerical solutions
of the non- Markovian model are presented. No assumption is made in the theoretical
derivation on the type of microscopic dynamics along the reaction coordinate.
However, the detailed calculations are for Brownian dynamics in which the velocities
are uncorrelated in time (but spatial memory remains).
Understanding thermoelastic martensitic transformations is a fundamental
component in the study of shape memory alloys. These transformations involve
a hysteretic change in stability of the crystal lattice between an austenite
(high symmetry) phase and a martensite (low symmetry) phase within a small temperature
range. To study these transformations, a set of phenomenological temperature-dependent
atomic pair-potentials is used to derive the crystal's energy density W(F,S1,
S2 , ... ;T) as a function of a uniform deformation F and a set of
internal atomic shift degrees of freedom S1 Special attention is
paid to the evaluation of crystal structure stability. Using a specific set
of temperature-dependent pair-potentials a stress-free bifurcation diagram is
generated for the B2 binary crystal structure (with temperature serving as the
loading parameter). A hysteretic transformation is suggested by the existence
of certain stable equilibrium branches corresponding to B2 (CsCl) and B19 (orthorhombic)
crystal structures. These results indicate the ability of temperature-dependent
atomic potential models to provide valuable insight into the behavior of shape
memory alloys such as NiTi, AuCd, and CuAlNi.
Jim Evans (Department
of Mathematics, Iowa State University)
From atomic scale ordering to mesoscale spatial patterns
in surface reactions: Heterogeneous coupled Lattice-Gas (HCLG) simulation approach
Slides: pdf
A challenge for the modeling of surface reaction-diffusion systems
is to connect-the-length-scales from a realistic atomistic treatment of local
ordering of reactants and reaction kinetics to an "exact'' description
of mesoscale spatial pattern formation. We discuss a heterogeneous coupled lattice-gas
(HCLG) approach which utilizes parallel kinetic Monte Carlo simulations of a
lattice-gas reaction model to simultaneously determine the local reaction kinetics
and diffusive transport properties at various macroscopic "points'' distributed
across the surface. These simulations are coupled to reflect macroscopic mass
transport via surface diffusion.
This method is demonstrated for bistable CO-oxidation reactions
on surfaces. We first discuss a simple model where surface transport of CO is
reduced from a many-particle to a single-particle problem (although one with
even more complexity than an "ant-in-the-labyrinth" type percolative diffusion
problem). However, we have also applied HCLG to analyze reaction front propagation
in a realistic model for CO-oxidation on Pd(100).
Ref. Da-Jiang Liu and JWE, PRB 70 (04) 193408; SIAM MMS 3 (05).
Jim Evans
(Department of Mathematics, Iowa State University)
Atomistic and continuum modeling strategies for homoepitaxial
thin film growth
Slides: pdf
Homoepitaxial thin film growth produces a rich variety of far-from-equilibrium
morphologies. Atomistic lattice-gas models analyzed by KMC simulation have been
most successful to date in predicting behavior observed in specific experiments.
However, 2D continuum formulations (level-set, phase-field, geometry-based-simulation
= GBS) retaining discrete layers have been explored as alternatives, especially
for the regime of highly reversible island formation where KMC becomes inefficient.
Exploiting GBS, we present the first precise results for the submonolayer island
size distribution in this regime [1].
3D continuum formulations have been applied to describe multilayer
kinetic roughening where step edge barriers inhibit downward transport and produce
unstable growth (mound formation). We analyze this phenomonon using realistic
atomistic modeling to show that Ag/Ag(100) [regarded as the prototype for smooth
growth] actually grows very rough [2]. Furthermore, mound dynamics is seen to
be more complex than predicted by standard 3D continuum models.
[1] PRB 68 (03) 121401; SIAM MMS 3 (05); [2] PRB 65 (02) 193407.
Gero Friesecke (University of Warwick
and Technical University Munich)
Interatomic 'van der Waals' forces and the Schroedinger
equation
Long range interatomic 'Van der Waals' forces play an important
role for equilibrium structure and nonequilibrium behaviour of complex molecular
systems (carbon nanotubes, DNA, proteins, ...), but must at present be modelled
empirically: ab initio computation remains out of reach. The latter would be
particularly desirable because of the huge chemical specificity (i.e., atom
dependence) of the VdW force, e.g. for a pair of sodium atoms it is bigger by
a factor 1000 than for two Helium atoms.
The difficulty is that 'order N' computational quantum models
like Hartree-Fock theory and density functional theory (and their variants integrated
into molecular dynamics in the spirit of Car-Parinello) do not resolve VdW forces,
only short range covalent bonding. No model short of full all-N-body Schroedionger
is known which captures VdW - but that is order eN (prohibitive).
Our main result is that the presence, magnitude and underlying
mechanism (quantum electron-electron correlations) of this attraction is in
fact a rigorous theorem about the many-body Schroedinger equation. This leads,
in particular, to an explicit expression at long range with greatly reduced
computational complexity: roughly speaking, enumber of electrons in
a single atom.
In the talk, we will start by reviewing the interesting history
(starting from van der Waals 1873) and the well developed chemistry 'lore' (starting
with Eisenschitz and London 1927) of van der Waals forces.
Matthias Kurzke (IMA, University of Minnesota)
Boundary vortices in thin magnetic
films
Poster: pdf
We analyze a model for thin ferromagnetic films that leads to
the formation of vortices at the boundary. The energy asymptotically splits
into a singular part depending only on the number of vortices and a finite part
depending on their position. This finite part, the renormalized energy, is shown
to also control the gradient flow motion associated to the boundary vortex functional.
The results and proofs are similar to the theory for Ginzburg-Landau vortices
by Bethuel-Brezis-Helein for the static and Sandier-Serfaty for the dynamic
case.
Claude Le Bris (Ecole Nationale des Ponts
et Chaussées (ENPC), CERMICS)
Discrete to continuum limit for various microscopic lattices: a
static picture
We overview some recent work, joint with xavier blanc and
pierre-louis lions, on the derivation of macroscopic densities
of mechanical
energy from energies of microscopic lattices. A particular
emphasis is laid on
the definition of the energy of some microscopic stochastic
lattices, with
possible applications to the modelling of materials.
Frederic Legoll (Institute
for Mathematics and its Applications, University of Minnesota)
Analysis of a prototypical multiscale method coupling
atomistic and continuum mechanics
In order to describe a solid which deforms smoothly in some region,
but non smoothly in some other region, many multiscale methods have been recently
proposed, that aim at coupling an atomistic model (discrete mechanics) with
a macroscopic model (continuum mechanics). We present here a theoretical analysis
for such a coupling in a one-dimensional setting. We study both the general
case of a convex energy and a specific example of a nonconvex energy, the Lennard-Jones
case.
In the latter situation, we prove that the discretization needs
to account in an adequate way for the coexistence of a discrete model and a
continuous one. Otherwise, spurious discretization effects may appear. We also
consider the effect of the finite element discretization of the continuum model
on the behaviour of the coupled model.
This work is joint with Xavier Blanc (Paris
6) and Claude Le Bris (CERMICS, ENPC).
Igor Mezic (Department
of Mechanical Engineering, University of California - Santa Barbara)
Dynamics and control of large-scale molecular motion
Finding rules for design of complex systems that have sufficient
flexibility to execute varied tasks while at the same time performing robustly
against a large class of perturbations is of considerable interest. Biological
networks are of particular interest in this context, given that they have those
desired features. We study a network of nearest-neighbor coupled oscillators
starting with a simple coarse-grained model of a molecule with a backbone and
side-chains. We show that this system is particularly good at reacting responsively
to localized disturbances by amplifying them. We present analysis of an interesting
transition phenomenon between global energy minima of the system and relate
this to recent results on controllability of Hamiltonian systems. More generally,
oscillator networks that exhibit such phenomena are characterized by nearest
neighbor interactions of a node that are, in most of the phase space, much stronger
than the nonlinear oscillations of the local dynamics at the node.
Alexander Mielke
(Institut für Mathematik, Humboldt-Universität zu Berlin)
Macroscopic equations for microscopic dynamics in periodic
crystals
Poster: pdf
ps
In infinite periodic lattices the solutions can be studied by
Fourier analysis on the associated dual torus. However, in doing a limit procedure
with vanishing atomic distance, one observes new phenoma which are usually studied
by WKB mehtods. We show that similar results can be obtained under much weaker
assumptions by using weak convergence methods.
First we show that linearized elastodynamics can be obtained
by a gamma limit procedure which automatically produces the effective elastic
tensor. Second we study the transport of energy in the lattice which occurs
on quite different wave speeds as the macroscopic elastic waves. It is possible
to derive a energy transport equation for a Wigner measure which depends on
time, space and the wave vector on the dual torus.
Alexander
Mielke (Institut für Mathematik, Humboldt-Universität zu Berlin)
Macroscopic dynamics in discrete lattices
Slides: pdf
We study effective macroscopic models for oscillations in oscillator
chains. We consider small-amplitude waves which are modulations of a basic periodic
pattern. If certain nonresonance copndtions hold and if the envelope function
satisfeis an associated nonlinear Schroedinger equation then the dynamics of
the oscillator chain remains a modulated wave which can be described by the
modulational theory on a suitably long time scale.
Similar results are observed for slow modulations of large-amplitude
traveling waves, where the macroscopic dynamics is now described by Witham's
modulation equation. However, rigorous proofs only exist in simplified cases.
Julie Mitchell (Mathematics and Biochemistry,
University of Wisconsin-Madison)
Computer prediction of protein docking and analysis of
binding interfaces
Slides: pdf
Recent work on the development of methods for protein docking
and analysis of binding interfaces will be discussed. One of the methods presented
is the Docking Mesh Evaluator that uses an implicit solvent model for electrostatics.
The Docking Mesh Evaluator is capable of exhaustive search as well as of local
and global optimization of binding energies, all of which can be performed using
parallel computation.
The Fast Atomic Density Evaluator is a method for analyzing protein
shape, and shape complementarity within binding interfaces. The Fast Atomic
Density Evaluator's complementarity "hot spots" correlate with residues in which
mutation is known to impact binding. This has recently been used in the development
of engineered ribonucleases able to kill cancer cells. Shape complementarity
analysis can also aid docking prediction, either as a post-filter for exhaustive
search results or as a means of dynamic parameterization for flexible docking
calculations.
Mirko
Hessel-von Molo
(Department of Mathematics, University of Paderborn)
Identification of macroscopic
dynamics
Several (combined) approaches for the
identification of macroscopic
(or essential) dynamical features of certain systems are
presented.
In particular the use of topological entropy with respect to
special
partitions
to identify almost invariant sets and "most expanding sets" of
a system
has been
investigated. Possible future work includes constructing a
Freidlin-Wentzell-like
quasipotential to construct simple models for high-dimensional
systems.
Jonathan A. Othmer
(Applied & Computational Mathematics, California Institute of Technology)
Coarse-grained phenomenological rate laws for nucleic
acid hybridization kinetics
Joint work with Justin S. Bois
and Niles A. Pierce (California Institute of
Technology).
Given a nucleic acid energy landscape defined in terms of secondary
structure microstates, we describe a coarse-graining approach for performing
kinetic simulations that accurately captures the temporal evolution of physically
meaningful macrostates. The method is based on the solution of local eigenvalue
problems that identify the dominant local relaxations between interacting macrostates.
The objective is the quantitative mapping of important landscape features for
functional nucleic acid devices.
Anja Riegert (Max-Planck Institute for the Physics of Complex Systems)
Modeling fast Hamiltonian chaos by suitable stochastic
processes
Projection operator techniques known from nonequilibrium statistical
mechanics are applied to eliminate fast chaotic degrees of freedom in a low-dimensional
Hamiltonian system. A perturbative approach, involving a Markov approximation,
yields a Fokker-Planck equation in the slow subspace which respects the conservation
of energy. A numerical and analytical analysis of suitable model systems demonstrates
the feasibility of obtaining the system specific drift and diffusion terms and
the accuracy of the stochastic approximation on all time scales. Non-Markovian
and non-Gaussian features of the fast variables are negligible.
Christof Schuette (Mathematics and Computer
Science, Freie Universität Berlin ) http://www.math.fu-berlin.de/~biocomp
Automated model reduction for complex molecular systems
A novel method for the identification of the most important metastable
states of a system with complicated dynamical behavior from time series information
will be presented. The novel approach represents the effective dynamics of the
full system by a Markov jump process between its metastable states / conformations,
and the dynamics within each of these metastable states by rather simple stochastic
differential equations (SDEs). Its algorithmic realization exploits the concept
of Hidden Markov Models (HMMs) with output behavior given by SDEs. The numerical
effort of the method is linear in the length of the given time series, and quadratic
in terms of the number of metastable states. The performance of the resulting
method is illustrated by numerical tests and by application to molecular dynamics
time series of a DNA oligomer.
Florian Theil
(Mathematics Institute, University of Warwick)
Crystallization in two
dimensions
Slides: pdf
Why do so many materials have a crystalline phase at low temperatures?
The simplest example where this fundamental phenomenon can be studied are pair
interaction energies of the type E(y)= 
0
< x < x' < N+1 V(|y(x)-y(x')|) where y(x) in R2 is the position
of particle x and V(r) in R is the pair-interaction energy of two particles
which are placed at distance r. We show rigorously that under natural assumptions
on the potential V the ground state energy per particle converges to an explicit
constant E*:
limN 
oo
1/N miny E(y) = E*,
where E* is the minimum of a simple function on [0, oo).
Furthermore, if suitable Dirichlet- or periodic boundary
conditions are
used, then the minimizers form a triangular lattice. To the
best knowledge
of the author this is the first result in the literature where
periodicity of ground states is established for a physically
relevant model which is invariant under the Euclidean symmetry
group
consisting of rotations and translations.
Giovanni Zanzotto
(Dipartimento di Metodi e Modelli Matematici per le Scienze
Applicate (DMMMSA),
University of Padua)
Stressed microstructures in M9R-M18R martensites
Joint work with Xavier
Balandraud (Laboratoire de Mécanique et
Ingénieries
(LaMI), Institut Français de Mécanique Avancée (IFMA),
Université Blaise
Pascal (UBP).
We revisit the phase transformation that produces monoclinic 'long-period
stacking' M9R or M18R martensites in Cu-based shape-memory alloys, and analyze
some associated microstructures, in particular a typical wedge-shaped configuration
(Fig.). The basic premise is that the cubic-to-monoclinic martensitic phase
change in such alloys is, geometrically, a slight modification of the well-known
bcc-to-9R transformation occurring for instance in Li and Na, whose basic strain,
at the micro level, is the same Bain strain as for the bcc-to-fcc transition.
One then determines the 'near-Bain' microstrain variants pertaining to these
elements and alloys, and analyze the long-period stacking martensite as a mesoscale
'adaptive phase.' Twins, habit planes, and also more complex microstructures,
such as the CuZnAl wedge, can be analyzed in this way. Earlier conclusions that
this microstructure is not kinematically compatible at zero stress are confirmed.
However, one can check the wedge is `close enough' to compatibility and compute
the corresponding stresses, which turn out to be low, causing only minimal plastification
and damage in the crystal. This microstructure is therefore rationalized as
a viable path for the transformation also in these alloys. One can moreover
verify this to be true for all the known lattice parameters reported for materials
exhibiting long-period M9R-M18R martensites. The general conclusion is that
the observed martensitic microstructures can be stressed to various degrees
also in good memory alloys; and that there seem to be no need for material tuning
in order tgat such stresses be low. Indeed, the lattice-parameter relations,
guaranteeing the zero-stress compatibility of special configurations, favoring
the transformation and its reversibility, do not need to be strictly enforced
because microstructural stresses are not very sensitive to lattice parameter
values.
