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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5908_Библиотеки_им_академика_М_И_Перельмана.pdf
Use of Molecular
Simulations to
8
Understand
Structural
Dynamics of
Antibodies
Daniel A. Nissley, Matthew I. J. Raybould,
Charlotte M. Deane, and Sandeep Kumar
8.1 WHY RUN MOLECULAR
SIMULATIONS ON ANTIBODIES?
Gaining insight into the structure‑function relationships that underlie antibody‑antigen
binding is a critical step in lead development. While static structures provided by com‑
putational predictions
9,10
phy
(MX) provide a wealth of information to direct, for example, mutational studies
to optimize antigen binding and limit off‑target interactions, they only ever provide
single snapshots of the ensemble of structures that an antibody populates in solution.
Furthermore, the MX structures themselves might contain deviations from the structure
that would be found in the solution state, as they are obtained under conditions that
are dissimilar to the environment they would encounter in a patient or in transport.9
1–8
or experimental methods like macromolecular crystallogra‑
201
202 Biopharmaceutical Informatics
Forexample, almost all MX structures are collected at cryogenic temperatures (~100 K)
to minimize radiation damage.11 Cryogenic electron microscopy (cryo‑EM) is another
experimental method of imaging biomolecules under more native‑like conditions.12
However, the low throughput of cryo‑EM experiments and a practical resolution thresh‑
old of ~3 Å13 currently limit their application in the drug development pipeline.
Diverse structural questions may be of interest during antibody drug development.
Frequently, these are related to the antibody‑antigen binding event: which of the con‑
tacts present in the MX structure remain when it is heated to room or body temperature?
Of those contacts that remain, which contribute most strongly to the overall free‑energy
change upon binding? Over time, how do the two binding arms reposition with respect
to one another in solution? How does a given mutation in a complementarity‑determin‑
ing region (CDR) loop alter the kinetics of binding? Each of these questions requires
dynamic information about antibody structures that is difcult or impossible to access
from the experimental structures alone.
Other questions may relate to the developability of the antibody, such as predicting
its propensity to aggregate or self‑associate, or its response to changes in salt content
made during formulation. These solution properties would ideally be assessed at an
early development stage before they could become problematic. Complicating the issue,
experimental and theoretical studies indicate that factors difcult to predict from either
sequence or structure alone, including the anisotropy of electrostatic interactions and
the characteristic oligomeric state of the antibody in solution, strongly inuence these
properties.
As described in the following sections, the careful application of molecular simula‑
tion techniques can provide suggestions or even answers to these and other common issues
that arise throughout the development of an antibody therapeutic. Before we consider these
case studies, we rst step through the most common molecular simulation techniques and
important considerations when designing a simulation study of antibody biophysics.
8.2 COMMON TYPES OF MOLECULAR SIMULATIONS FOR BIOMOLECULES
8.2.1 Molecular Dynamics (MD) Simulations
For biomolecules, molecular dynamics (MD) simulations are perhaps the most widely
used molecular simulation technique today. MD is a mature, broadly used technique
as recognized by the 2013 Nobel Prize in Chemistry awarded to Karplus, Levitt, and
Warshel for the development of multi‑scale modeling techniques for complex chemical
systems, including MD. In this section, we describe in general terms the theory of MD
simulations.
The main goal of MD is the prediction of the time evolution of a molecular sys‑
14,15
In our application, the system likely consists of an antibody and perhaps its
tem.
antigen in aqueous solution in a patient. We choose to model the system using the rules
8 • Antibody Structural Dynamics 203
T
E EX
()
=
E
bonded
E
non‑bonded
E EE=+
bonded non‑bonded
E EEE=++
bonded bonds angles torsions
E EE=+
non‑bonded VDW el
E
bonded
E
non‑bonded
FE=−∇
of classical (Newtonian) mechanics and represent each particle as a sphere and each
bond as a spring. At the beginning of the simulation, each particle in the system is ini‑
tialized with some starting position and velocity. Initial particle velocities are often ran‑
domly selected from a Maxwell‑Boltzmann distribution generated at the user‑dened
absolute temperature of the system,
. A mathematical function, termed a forceeld or
Hamiltonian, E, determines the forces different particles in the simulation exert on one
another. Despite their name, forceelds are typically written as expressions that, given
system particle coordinates X, compute the potential energy of the system
The specic terms within the forceeld are chosen to match quantum‑mechanical cal‑
culations and, in most cases, experimental data. Typically, potential energy expres‑
sions contain terms that constrain particle motions through both bonded terms,
representing constraints on particle motions due to bonds, angles, and torsions, and
non‑bonded terms,
, representing non‑covalent interactions like Van der
Waals (VDW) and electrostatics:
.
,
Equations (8.2 and 8.3) provide a typical decomposition of the
and
(8.1)
(8.2)
(8.3)
terms in Equation (8.1) into their component parts. Some typical mathematical func‑
tions used to represent bonds, angles, torsion angles, VDW, and electrostatic interac‑
tions are shown in Figure8.1a–e.
With positions and velocities determined, the forces acting on particles are then
computed as the negative gradient,
, of the potential energy. Having deter‑
mined the forces, Newton’s equations of motion are then used to predict the positions
and velocities at some future time of the system. To reduce numerical integration
errors, the time step must be small; values on the order of 1 to 10 femtoseconds are
typical for MD simulations of biomolecules. By stepping through time and updating
the positions and velocities of all particles in the system according to the forceeld,
we produce a movie in 3D space of the time evolution of the system. Various meth‑
ods for maintaining the system temperature during the simulation by updating or
rescaling particle velocities have been developed.
16–19
Analysis of the resulting “tra‑
jectory” of the system based on the coordinates collected during the run can then be
carried out.
Several different types of MD are frequently encountered in the academic liter‑
ature, all of which fall under the umbrella of MD techniques. In Langevin dynam‑
ics,18additional drag and random collision terms are added to Newton’s equations of
motion to approximate the inuence of solvent. In Brownian Dynamics,20 which may be
considered a simplied form of Langevin dynamics that applies to certain particles in
solution, the equations of motion are modied such that there is no average acceleration
in the system.
204 Biopharmaceutical Informatics
FIGURE8.1 Common mathematical forms of MD forceeld terms are shown in panels
A through E.
87
Each equation is accompanied by a plot generated with sample forceeld
parameters.
(Continued)
8 • Antibody Structural Dynamics 205
E
bond
k
b
r
0
k
kcal
rÅ
0
=
k
θ
0
k
kcal
mol
600=°
E
torsion
k
ϕ
k
kcal
mol
=π
n 1, 2,3
{}
=
6
=
kcal
mol
i
r
ij
i
1=+
j
1=+
i
1=+
j
1=−
(Continued)
(a) In the expression for
and equilibrium bond lengths. The plot shows results for a bond with
and
. (b) Angles are computed as a function of the force constant
2
,
is a force constant and r and
are, respectively, the current
50
=
b
×
Å
mol
and the squared
2
deviation between the equilibrium (
0.1
=
is shown for
, the multiplicity n (equal to the integer number of minima in the period
,
radians, and
0.1
=
θ
, and its phase
tion of
force constant
[−π, π]), the torsion angle
ϕ
) and current () bond angles. The energy as a func-
and
. (c)
is expressed as a function of the
. The displayed plots were generated using
. (d) The Lennard-Jones potential or a slightly
modied form is frequently used for VDW interactions; it computes the non-bonded interaction between two particles as a function of the distance between the particles, r, the
collision diameter for the interaction,
displayed for
Å
and
5
=
Coulombic term; the charges of particles
the permittivity of vacuum, and
are shown for
and
, and the depth of the potential well, . Results are
. (e) Electrostatic interactions may be treated with a
and j are given by i and j, respectively, 0 is
is the current distance between particles i and j. Results
(blue) and for
and
(green). (f) (Top) Cartoon
model of an all-atom antibody Fab in a periodic simulation box. (Bottom) All-atom and CG
models of the same Fab are shown. In the CG model, each amino acid is reduced to one
representative interaction site.
8.2.2 Monte Carlo (MC) Simulations
Monte Carlo (MC) simulation refers to a broad range of simulation tools in which ran‑
dom numbers are used to perform a search across some phase space.18 The goal of an
MC simulation is typically the same as an MD simulation: the prediction of a quantity
of interest from the system. In the case of biomolecules, we are typically concerned
with exploring the conformational and energetic landscape of our system of interest to
predict its conformations or energies. In many cases, a forceeld much like those used in
MD simulations will be used to assess the energy of molecular conformations.
MC and MD simulations may also be used within the same protocol, such as in the
enhanced sampling technique of replica exchange.21 In replica exchange, a set of identi‑
cal replicas of a chemical system are initialized, each at a different temperature. After a
short MD simulation, exchanges are attempted between “neighboring” replicas that are
at adjacent temperatures. The probability that the exchange is accepted, w, is calculated
based on the energy and temperature differences between the two replica conforma‑
tions, and a random number,
is accepted and the replicas are swapped between temperatures. By performing many
thousands of these exchanges, the replicas undergo a random walk in temperature space.
This random walk helps the simulation avoid spending most of its runtime in local
potential energy minima and thereby reduces the total time required to fully explore the
energy landscape relative to a single‑temperature simulation.
of the multitude of MC methods that may be applied to molecular simulations.
, is generated on the interval [0,1]. If
22
This is but one example
, the exchange
206 Biopharmaceutical Informatics
We note that the key difference between MC and MD simulations is that subsequent
structures generated by an MC method are not time‑correlated with one another as they
are in an MD trajectory.
8.2.3 Challenges of Molecular Simulations
As with all models, molecular simulations have certain limitations that must be consid‑
ered. As described above, the goal of molecular simulation is normally to predict the
time evolution or ensemble average of some property for a chemical system of inter‑
est. Due to the use of random numbers when generating particle velocities (as well
as numerous other technical factors23), individual MD trajectories, even when initiated
from identical starting coordinates, quickly diverge from one another. The use of ran‑
dom numbers in MC simulations likewise leads to divergent behavior between runs.
This leads to questions such as: which of these simulations can be considered correct?
Are they all reasonable predictions of behavior?
The random nature of the simulations means that each can be thought of as simi‑
lar to one observation from a single‑molecule experiment–each is individually valid,
but the average (ensemble) behavior of the system only becomes clear in the limit
of many simulations/observations. In practice, this means that multiple simulations
are run and the conformations sampled across them are averaged together. If we are
trying to compute the average radius of gyration, we compute the radius of gyration
of all of the different conformations of the molecule from each simulation and aver‑
age them. Determining if enough statistically independent simulations of sufcient
length have been run to collect a representative sample of possible conformations is
a key issue in the practical application of molecular simulations. Obtaining sufcient
sampling over the different possible conformational states/trajectories of the system
to achieve converged results is time consuming and expensive, especially for larger
systems.23 Some systems are simply too large to achieve converged results,24mean‑
ing that reliable answers cannot be obtained without reducing the complexity of the
calculation. Below we describe methods that have been developed to overcome these
sampling problems.
Molecular simulations also require starting structures. In most cases, these are MX
structures deposited in the Protein Data Bank25 (PDB). However, many experimental
structures have residues with missing side chains or sections in which entire residues
could not be resolved, which require careful rebuilding before simulation. Most pro‑
teins solved by MX are small globular proteins or single domains of multi‑domain
proteins crystallized in isolation. This means that, in many cases, multiple PDB models
must be merged to build a complete model for simulation. With the advent of more
accurate structure prediction tools such as AlphaFold
protein structure prediction tools,
4–8
the reliance on experimental structures is begin‑
ning to ease. This explosion in predicted structures has opened up exciting new oppor‑
tunities for simulations by providing a wealth of starting structures for previously
inaccessible systems.
One shortcoming of classical MD simulations is that they cannot model the for‑
mation and breakage of chemical bonds, meaning that they cannot be used to model
1–3
and various antibody‑specic
8 • Antibody Structural Dynamics 207
enzymatic reactions. Several methods are available to overcome this shortcoming.
Multi‑scale modeling methods in which most of the biomolecular system is represented
classically and the catalytic region is modeled using quantum‑mechanical methods have
been developed,26 and some specialized forceelds can model bond breakage and for‑
mation.27 However, in simulations of antibodies, we tend to be interested in either anti‑
body‑antibody or antibody‑antigen non‑covalent interactions, meaning that the majority
of simulations run are classical.
8.3 MODELING PERSPECTIVE: WHY
WE CANNOT SIMULATE EVERYTHING
IN THE REAL SYSTEM
The goal of a molecular simulation is to predict a property of interest for an antibody.
In practice, this is achieved by initializing a simulation with some set of particles rep‑
resenting the molecules in the system, dening how they interact with one another, and
then by some method sampling the different accessible conformational states of the
simulation.
It makes intuitive sense that the most accurate simulation would be one in which
every molecule present in the real system and their interactions with one another are
represented as accurately as possible. However, the cellular or test tube environment is
far too large and complex to be explicitly simulated with current computational power.
For example, a 1‑mL aliquot of a 150‑mg/mL full‑length immunoglobulin G 1 (IgG1)
monoclonal antibody (mAb) solution contains on the order of 1017 antibody molecules
(assuming a molecular weight of 150 kDa28), each of which is composed of ~20,000
atoms. Without even considering the need to add water and cosolutes to our simulation,
we can see that the number of atoms exceeds 1021. Complete simulations of a cell‑like
environment would be necessarily even more complex, containing each of the thou‑
sands of unique macromolecules and small molecules composing the cellular milieu.
Current computational power limits molecular simulations to an upper bound of 109
particles, though this feat required utilizing 65,000 processors.29 Simplifying assump‑
tions must be applied to reduce the size and complexity of the real system by many orders
of magnitude to make up the difference between the ≤109‑atom systems we can simulate
and the real system of 1 mL of 150‑mg/mL mAb with >>1021 atoms that we cannot. Some
of the most important and frequently employed simplifying assumptions are considered
below before we discuss different resolution molecular simulations in detail.
8.3.1 Periodic Boundary Conditions
To avoid needing to explicitly model large volumes of solution, molecular simulations
typically apply periodic boundary conditions around a single copy of the biomolecule of
interest. These boundary conditions dene an innitely mirrored simulation space, such
208 Biopharmaceutical Informatics
that when a particle’s trajectory causes it to exit the system from one end, it reappears
at the other end with the same velocity (Figure8.1f). The volume within the periodic
boundary cell is chosen to be sufciently large that the protein cannot interact with itself
through the periodic wall during the simulation. Thus, these simulations can be thought
of as being run at innite dilution. Limiting the size of the simulation in this way mas‑
sively decreases the number of particles to within the realm of feasibility.
8.3.2 Inclusion versus Exclusion of
Constant Domains
Structurally, antibodies can be broken into the Fab regions at the ends of the two bind‑
ing arms that recognize and bind antigens and the Fc region that is involved in signal‑
ing. Due to the strong emphasis on understanding antibody‑antigen binding, as well
as the inherent difculty in crystallizing full‑length antibodies (and larger proteins
in general), experimental structures predominantly contain only Fab segments. For
example, as of the 1‑Nov‑2022 update, the SAbDab database
tures contains 6,029 Fvs, 5,084 Fabs, and 17 full‑length antibodies (all sequence
non‑redundant). These numbers indicate that while the entire Fab is present in 84% of
structures, intact antibodies are rarely crystallized. Molecular simulation studies are
frequently performed using only the Fv and antigen to reduce computational expense.
This approximation appears to make intuitive sense for cases in which only the ener‑
getics of antigen/antibody interactions are of interest, as the entire interface is resolved.
However, published MD simulations indicate that including the full Fab rather than just
the Fv does, in fact, inuence results,32 suggesting that using full Fabs whenever pos‑
sible in simulations should be considered a matter of best practice. Simulations of the
properties of solutions of mAbs, however, require a representation of the Fc to account
for, at a minimum, the effects of its steric bulk. Simulations of full‑length antibodies
have been published,
section of the molecule.
33–38
though they are far less common than studies using a reduced
30,31
of antibody struc‑
8.3.3 All‑Atom versus Coarse‑Grain (CG)
Simulations
Simulations in which each atom of the chosen reduced chemical system is explicitly
represented are the most common type of molecular simulations performed. All‑atom
simulations of this type must use notably short integration time steps of 1 or 2 fs in order
to maintain stability during numerical integration. The magnitude of the integration
time step is limited by the highest‑frequency vibrations in the system, which for atomic
systems described classically are the bond vibrations.39 By constraining covalent bonds
containing hydrogen atoms, a time step of up to 2–3 fs may be used. All‑atom methods
are considered the standard for accuracy in MD simulations. All‑atom MD simula‑
tions also benet from a plethora of well‑used and documented tools for preparing,
running, and analyzing simulations, including Amber,
OpenMM,43 and NAMD.44 The popular forceelds like Amber also have many iterations
40
GROMACS,41 CHARMM,42
8 • Antibody Structural Dynamics 209
that are best applied in different situations, making the choice of forceeld an important
question. A benet of all‑atom methods is that all of the mainstream protein force‑
elds (e.g., CHARMM, Amber) are transferable, meaning that they can be applied to
any protein system without needing to generate custom parameters. As we will see
below, reduced‑resolution models frequently include non‑transferable terms that must
be parameterized on a case‑by‑case basis. All‑atom simulations are typically run with
explicit representations of solvent molecules and ions. Various water models exist, and
some are designed to work in concert with specic biomolecular forceelds (for exam‑
ple, the Amber FF14SB protein forceeld45 is optimized to run simulations with the
TIP3P water model46).
Reducing the computational expense of all‑atom simulations to allow the simula‑
tion of larger systems for longer timescales while preserving as much of their predictive
power as possible is highly desirable. One method to reduce the computational cost
of MD simulations is to coarse‑grain (CG) the system by reducing groups of atoms to
representative interaction sites
47,4 8
(Figure8.1f). In most cases, this is achieved by start‑
ing with an atomistic structure of the system and using a CG mapping function that
determines how groups of atoms are reduced to interaction sites.48 Once a CG mapping
has been determined, a forceeld that describes the interactions between CG sites can
be used to calculate the forces between them and their time evolution is then predicted
with Newton’s equations of motion, just as in all‑atom MD.
Simulations of CG models are faster than all‑atom simulations for several reasons.
First, by reducing the number of particles in the system, they signicantly reduce the
number of forceeld terms that must be computed at every time step. Returning to our
example of a ~20,000‑atom full‑length IgG1mAb, if we CG it to one interaction site per
residue, we reduce the number of particles by an order of magnitude to ~1,400 interac‑
tion sites. Reducing the number of degrees of freedom in the system not only reduces the
number of forceeld terms to compute but also reduces the roughness of the free‑energy
landscape,47 tending to accelerate dynamic processes like protein folding. By increas‑
ing the length scale of the system and reducing the frequency of the highest‑frequency
vibrations, CG models can also typically be run with integration timesteps up to 10‑fold
larger than all‑atom MD, allowing them to take larger leaps through time and simulate
longer timescales faster. Finally, CG models are frequently run without explicit solvent
representations, opting instead to represent the solvent implicitly to further reduce the
number of particles. Together, these effects mean that CG simulations are often several
orders of magnitude faster than all‑atom simulations.
The selection of model resolution is a critical modeling decision that is typically
made at the earliest steps of a molecular simulation project. In general, the selection
of model resolution should be motivated by the time and length scales of the process
of interest. In fact, it is always worth considering whether the time and length scales of
the property of interest put it outside of the practical realm of MD simulations. In terms
of timescale, all‑atom simulations can routinely access timescales on the order of 10−9
to 10−6 seconds (nano‑ to microsecond), with longer simulations (up to millisecond)
possible for small systems or with the heroic application of computational power. For
43,49
example, the benchmarks
for OpenMM v7.7 on a single A100 GPU indicate that for
a ~105‑atom system (apolipoprotein A1) one can expect to achieve 429 ns of simulation
per day, but with a ~106‑atom system (satellite tobacco mosaic virus), that performance
drops to just 32 ns/day on the same hardware. At the far upper limit of the computational
210 Biopharmaceutical Informatics
performance curve, the state‑of‑the‑art Anton 3 supercomputer, which is custom‑built
for high‑speed MD simulations, boasts a speed of 100,000 ns/day using 512 nodes to
simulate a 106‑atom system.
The speed‑up of a CG simulation relative to an equivalent all‑atom system is not
frequently reported in the literature. General estimates suggest CG models tend to be
103 to 104 times more efcient than all‑atom simulations.51 In some cases, however,
the improvement can be more drastic; a set of 19 CG models of globular proteins were
found to fold on average 4 × 106 times faster than in experiments.52 The specic sources
of acceleration in CG simulations are many and differ between models; we direct inter‑
ested readers to the discussion in Refs. 47 and 48.
While CG models enjoy favorable increases in simulation speed, they suffer from
a loss of spatial resolution. CG models with one interaction site per amino acid placed
at the coordinates of the Cα atom are reduced to a spatial resolution on the order of the
average Cα–Cα bond length, 3.8 Å. In many cases when simulating mAbs, resolution
is reduced much further, with the mAb represented by perhaps 6, 10, or 12 interac‑
tion sites. These simulations, which may be considered “ultra‑coarse‑grained” as they
reduce hundreds of amino acids to single interaction sites, can be used to simulate
protein‑protein interactions and predict experimental solution properties. However, a
six‑bead model of antibodies is not suitable for the investigation of which residues
contribute to antibody‑antigen binding as the individual residues at the interface are
not modeled. Given that an innite number of different CG representations may be
designed for a given protein, methods to determine the “best” representation have
emerged.53 However, in most cases, researchers employ a combination of intuition for
the chemical system under investigation and trial and error to achieve a realistic model
for the system and parameter of interest. Despite these issues that would appear to
limit the accuracy of CG models, when carefully parameterized and used within their
limitations, they can accurately predict the dependence of protein properties on osmo‑
lyte concentration,54 the inuence of pH on antibody viscosity,55 and various other
properties.
56,57
50
8.4 USES OF MOLECULAR SIMULATION IN ANTIBODY DRUG DEVELOPMENT
8.4.1 Predicting and Understanding Protein‑Protein
Interactions in mAb Solutions
Most antibody drugs, such as mAbs, are delivered by sub‑cutaneous injection. The com‑
fort of the patient receiving the injection places certain limits on the volume (1–2 mL58)
that can be administered and the viscosity (practical delivery threshold ~20 cP
solution, the latter of which determines the injection force required. The injection volume
limit and typical dosages dictate mAb solutions with concentrations of >100 mg/mL.58
59
) of the