# Selecting for adaptability in materials design

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Design principles for adaptability revealed by time-varying optimization

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Design principles for adaptability revealed by transient optimization

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Design principles for adaptable materials revealed by incomplete
optimization

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Design principles for adaptable materials revealed by incomplete
optimization

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Design principles for adaptability revealed by incomplete optimization

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Adaptable materials through incomplete design

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Adaptable materials revealed by repeated incomplete design

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Adaptable materials revealed by repeated incomplete design

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Non-equilibrium training protocols for adaptable materials

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

# Learning to learn: Non-equilibrium design protocols for adaptable materials

Martin J. Falk  Department of Physics, The University of Chicago, Chicago, IL
60637  Jiayi Wu  Department of Physics, The University of Chicago, Chicago, IL
60637  Ayanna Matthews  Graduate Program in Biophysical Sciences, The
University of Chicago, Chicago, IL 60637  Vedant Sachdeva  Graduate Program in
Biophysical Sciences, The University of Chicago, Chicago, IL 60637  Nidhi
Pashine  School of Engineering and Applied Science, Yale University, New
Haven, CT 06511  Margaret Gardel  Department of Physics, The University of
Chicago, Chicago, IL 60637  James Franck Institute, The University of Chicago,
Chicago, IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago, IL 60637  Pritzker School of Molecular Engineering, The
University of Chicago, Chicago, IL 60637  Sidney Nagel  Department of Physics,
The University of Chicago, Chicago, IL 60637  James Franck Institute, The
University of Chicago, Chicago, IL 60637  Arvind Murugan  Department of
Physics, The University of Chicago, Chicago, IL 60637  James Franck Institute,
The University of Chicago, Chicago, IL 60637

###### Abstract

Evolution in time-varying environments naturally leads to adaptable biological
systems that can easily switch functionalities. Advances in the synthesis of
environmentally-responsive materials therefore open up the possibility of
creating a wide range of synthetic materials which can also be trained for
adaptability. We consider high-dimensional inverse problems for materials
where any particular functionality can be realized by numerous equivalent
choices of design parameters. By periodically switching targets in a given
design algorithm, we can teach a material to perform incompatible
functionalities with minimal changes in design parameters. We exhibit this
learning strategy for adaptability in two simulated settings: elastic networks
that are designed to switch deformation modes with minimal bond changes; and
heteropolymers whose folding pathway selections are controlled by a minimal
set of monomer affinities. The resulting designs can reveal physical
principles, such as nucleation-controlled folding, that enable adaptability.

Considered as materials, biological systems are striking in their ability to
perform many individually demanding tasks in contexts that can often change
over time. This success can be attributed to “meta-properties” like
modularitykashtan2005spontaneous ; Parter2008-ez ; hemery2015evolution ;
kashtan2007varying ; lipson2002origin , robustnesswagner2008robustness ,
plasticity for learninggupta2021embodied , and multifunctionalitydu2020inverse
; rocks2019limits ; sachdeva2020tuning ; Murugan2021-eg . While inverse
materials design has sought to optimize specific properties
rechtsman2006designed ; Zeravcic2014-yb ; Coli2022-qy ; Zunger2018-wp ;
Murugan2019-ls ; Lipson2000-ar ; Bianchi2012-an ; Engel2015-fs ; Zhang2005-wt
; Bianchi2012-an , less attention has been given to identifying general design
strategies for creating materials with meta-properties.

Here, we show how a biologically-inspired design method can target one such
meta-property, adaptability. By adaptability, we mean the ability to switch
between mutually-incompatible functions with minimal changes in design
parameters. For example, an adaptable elastic network could switch from a
negative Poisson ratio to a positive one with minimal network changes, even
though a single network alone can only have one Poisson ratio. In this
example, the mutually-incompatible functions are the different Poisson ratios,
and the design parameters are the stiffnesses of the network bonds. A truly
adaptable material will be as good as a non-adaptable material at any given
function, but will require fewer modifications to produce a distinct,
incompatible function.

At first glance, the existence of a truly adaptable material seems highly
improbable. However, if the design space of the material is high-dimensional,
then we should generically expect that there are many distinct choices of
design parameters with equivalent performance for a given
functionMurugan2019-ls ; kashtan2005spontaneous ; Greenbury2022-sd ;
Van_Nimwegen2006-yz ; Ebner2001-rs ; rocks2020revealing ; rocks2021hidden .
Our goal is to identify the much more rare subsets of design solutions which
both perform the given function and are adaptable.

Figure 1: Evolving design parameters towards different target functionalities
over time can select for highly adaptable materials. (a) Materials can be
characterized by their design parameters (e.g. for an elastic network, the
rest length and stiffness of each bond are design parameters). The design
parameter-to-function map is often highly degenerate; many choices of design
parameters (e.g., orange region) can achieve the same equivalent function
(e.g., $G2$). Some parts of orange region might be close to design parameters
(green) that achieve an alternative function $G2$; such regions can often be
vanishingly small but correspond to materials that can quickly adapt from
exhibiting function $G1$ to $G2$ with minimal changes. We have drawn regions
for each goal as simply connected regions, but the topology may be more
complex. (b) Optimization for a single, fixed goal either $G1$ or $G2$ will
typically result in non-adaptable materials that are good at function $G1$ or
$G2$ but not adaptable. Switching between the two goals on a fixed timescale
$\tau$, even if the parameters are not yet well-adapted to the current target
goal, by construction selects for parameters that are closer together in
parameter space. Faster switching selects for closer parameter sets that might
be rarer or might not exist.

In our approach, we take existing optimization algorithms for a target
function and repeatedly switch the target before optimization is completed for
any one function. The partially adapted design parameters for one function are
used as initial conditions for optimizing the second function. This
intuitively requires the solutions identified in successful periods of
training to drift closer to each other in design space with each switch (Fig.
1).

The underlying logic of this approach is that the sets of design parameters
which survive the oscillating selection process are required to be similar by
construction. This implies that there are shared design characteristics
between the solutions, even though the functions they perform are
incompatible.

We note that our approach does not directly optimize a metric of adaptability.
Instead, material adaptability arises because of the sequence of selection
pressures the material is subject to during design optimization. Our method
functions as a wrapper to existing optimization algorithms. It is therefore
compatible with a wide range of pre-existing materials design procedures,
ranging from fully computer-basedrechtsman2006designed ; Coli2022-qy  to fully
in situpashine2019directed ; pashine2021local .

Our work extends intuition developed earlier on
modularitykashtan2005spontaneous ; hemery2015evolution  in biological contexts
to canonical synthetic materials platforms. However, in these prior works, the
‘environment’ itself was chosen to be modular. The resulting system reflected
the modules specified by the environment: e.g., with logic
circuitskashtan2005spontaneous , the environment switched between selecting
for computing an AND operation between two sub-goals, or an OR operation
between those same two sub-goals. Consequently, the resulting logic circuits
developed modules for computing these sub-goals which could be quickly
recombined to achieve AND or OR with minimal changes to the circuit.

In contrast, in many problems relevant to materials, the different goals or
functions required may not have any obvious modular structure. For example,
consider two goals $G_{1},G_{2}$ representing a material with different
Poisson ratios, or a polymer folding into two distinct structures with no
common sub-structures. In this work, we focus on such arbitrary goals that are
not chosen to be modular in any obvious sense. We will nevertheless use the
alternating selection paradigm of prior biological works; we find that such
design protocols can _reveal_ adaptable organization of materials that can be
rationalized in retrospect, even for goals not organized in any obvious
modular manner.

We demonstrate the utility of this method in the context of two distinct
simulated systems - (a) soft materials with locally tunable elastic modulii,
and (b) self-assembling heteropolymers with tunable interaction affinities
between monomers. In both systems, rapidly oscillating training goals allows
us to find design parameters which can switch between mutually exclusive
functions with minimal parameter changes. In the self-assembly case, we gain
physical insight into the origin of adaptability, as selecting for
adaptability localizes parameter changes to interaction units which control
kinetic barriers in the folding landscape. Similarly, in the elastic networks,
we find that adaptability arises from a coherent displacement unit which is
easily shifted to perform opposing allosteric motions. Thus, our work suggests
a broad strategy to discover physical mechanisms that enable adaptability in
materials with arbitrary underlying physics.

## Results

### Mechanical metamaterials

Figure 2: Oscillatory training protocols generate adaptable solutions for
allostery in elastic networks (A) We perform training on elastic networks with
tunable bond stiffnesses. Goals $G1$ and $G2$ seek extensile and contractile
strains respectively at a pair of target nodes in response to strain at a pair
of input nodes. Target nodes, applied strain, and input nodes are the same for
$G1$ and $G2$. (B) Each successful run of training generates a pair of elastic
networks; one performs $G1$, the other performs $G2$. We show the network
which performs $G1$, bonds colored by relative stiffness. Bond stiffnesses
which change $>10\%$ to switch to $G2$ circled in black. Network examples from
training without oscillations (left), and with oscillations (right). (C)
Performance on each goal $G1,G2$ quantified by a cost function. Lower cost
function indicates improved performance. Training drives cost function of on-
target goal lower during each period. Background color panels indicate on-
target goal. $\tau$ indicates training steps per period of goal oscillation.
(D) Faster oscillation (smaller $\tau$) during training gives networks with
higher adaptability (defined as fraction of all bond stiffnesses which change
beyond $10\%$ threshold when switching between $G1$ and $G2$). Violin plots
show distribution of changed bond fraction over successfully trained network
pairs, black lines indicate minimum, mean, and maximum values.

In the context of mechanical materials, we first focus on allosteric response,
that is, the ability of an elastic network to exhibit a desired strain at a
distant target site if strained at a specific source site. Such allosteric
responses have been created through multiple methods rocks2017designing ;
yan2017architecture ; yan2018principles ; tlusty2017physical ;
flechsig2017design ; hemery2015evolution . Typically, there are multiple
design parameters that give rise to an allosteric response, which means that
the design space is degenerate with respect to that particular allosteric
motion. Our goal is to search through this degenerate design space for the
smaller set of parameters that can perform a specific, different, incompatible
allosteric response with minimal adjustment.

We use a 2-d mass-spring network to model allostery in a mechanical system.
Specifically, we simulate a 22-node hexagonal lattice with fixed boundary
conditions. While the geometry and rest lengths of all springs are fixed, the
collection of 83 spring constants $K=\\{k_{i}\\}_{i=1}^{83}$ are design
parameters that can be tuned to get specific allosteric responses. The two
motions we train for, $G1$ and $G2$, correspond to two opposite responses for
the same input at the same source site (Fig. 2A). We first reproduce such
single-function design by starting from many random initial conditions and
running a gradient descent procedure on a fitness function related to the
dynamical matrix.

Our fitness function rewards the softest mode transmitting strain from source
to target while also rewarding a large energy gap between such a soft mode and
the next mode; see Supplementary Information for further detail. Other works
have created such allosteric materials using a range of different fitness
functions and optimization proceduresrocks2017designing ; yan2017architecture
. Our procedure, while different in detail from previous work, nevertheless
consistently produces networks with the desired allosteric response.

Crucially, we find that there are numerous choices of design parameters $K$ \-
here, bond stiffnesses - that can separately perform each of the goals $G1$
and $G2$. One of these networks for $G1$ is shown in Fig. 2B(left); the others
are very different in the choice of $K$ but are equivalent in terms of
performance at $G1$.

Figure 3: Oscillatory training protocols generate networks with adaptable
Poisson ratios. (A) 2D elastic networks with disordered geometry are trained
for Poisson ratios $\nu>0.75$ ($G1$) or $\nu<-0.75$ ($G2$). During training,
bonds are irreversibly removed based on local stress in response to applied
strain. (B) Each successful run of training generates a pair of elastic
networks; one network performs $G1$, the other performs $G2$. We show the
network which performs $G1$. Bond changes required for switching to $G2$ are
circled in black. Network examples from training without oscillations (left),
and with oscillations (right). (C) An ensemble of $200$ networks undergoing
oscillatory training with bond removal results in successful switching between
$G1$ and $G2$ in 12 networks. Background color panels indicate on-target goal.
$\tau$ indicates bonds removed per period of goal oscillation. (D) Successful
networks trained with oscillation have a bond difference of $20$ bonds.
Networks trained without oscillation showed bond differences ranging from
$50-100$.

To leverage this degeneracy in design parameters for adaptability, we studied
a family of algorithms in which the target goal is switched periodically
between $G1$ and $G2$ during design optimization at different timescales
$\tau$, where $\tau$ is the number of optimization steps per period. In this
way, design parameters partially optimized for one goal, say $G1$, are used as
initial conditions for the next period of design that targets $G2$. Following
our intuition in Fig. 1, any two sets of design parameters produced
consecutively in this way are likely to be similar.

The results of switching at different frequencies are shown in Fig. 2B-D. We
find that such oscillatory design has two distinct phases. Initially, the
design parameters are not good at either goal $G1,G2$ at the time of training
goal switches (Fig. 2C). After this phase, we find one of two outcomes: (a)
success, i.e., convergence to a limit cycle between a pair of design
parameters (bond stiffnesses) $K_{1}$ and $K_{2}$ that are good at $G1$ and
$G2$ respectively (Fig. 2C) or (b) failure, i.e., convergence to design
parameters that are good at neither property.

In successful cases, we can measure adaptability of the pair $K_{1},K_{2}$ as
the number of bonds that need to change their elastic constant $k_{ij}$ by
more than $10\%$.

To understand the robustness of our procedure, we repeated the above
simulations for $500$ random initial assignments of bond stiffness for each of
$4$ different oscillation timescales $\tau$ (Fig. 2D). We see a substantial
and systematic increase in adaptability with frequency of switching, when
restricting to successful runs. Intuitively, for more rapid switching times,
when the process does converge on a limit cycle, the pair $K_{1},K_{2}$ are
closer, as they must be since there is less design time to get from one to
another.

For example, using an oscillation timescale $\tau=50$, we discover networks
that can switch function by changing as few as $5$ bonds (Fig. 2B(right)). In
contrast, networks obtained by optimizing for $G1$ or $G2$ alone typically
differ significantly in $40$ bonds (Fig. 2B(right),D).

To better understand how adaptability arises under oscillatory training, we
optimized several larger networks. While in small networks it was more
difficult to identify physical principles in the optimized networks, in larger
networks the mechanical signatures of allostery were visually clear. In one
example network, we found that oscillatory training produced a section of the
network which moved coherently (Supplemental Fig. 1, bottom). This section is
diverted with just a small number of bond stiffness changes, shifting from an
in-phase to an out-of-phase motion. Surprisingly, training for only one motion
at a time also produced networks with coherent motions (Supplemental Fig. 1,
top). However, the coherent motions selected for in the non-oscillatory
training differed between the $G1$ and $G2$ goals. This contrast between
oscillatory and non-oscillatory pairs was observed in several other large
networks (Supplemental Fig. 2).

#### Local learning rules

Figure 4: Oscillatory training protocols generate adaptable solutions for
heteropolymer folding. (A) We train the monomer interaction matrix for a
heteropolymer of length $13$ in order to target different folded structures -
a clockwise spiral $G1$ and a counter-clockwise spiral $G2$ \- in finite time,
starting from an unfolded state. Monomers are colored according to position.
(B) Performance on each goal $G1,G2$ is quantified by a cost function. Lower
cost function indicates improved performance. Training drives cost function of
on-target goal lower during each period. Background color panels indicate on-
target goal. $\tau$ indicates training steps per period of goal oscillation.
(C) Trained interaction matrices that target a spiral $G1$ and an anti-spiral
$G2$: (bottom) matrices $A_{1}$, $A_{2}$; upper triangle is the matrix, and
the lower triangle shows upper quartile interaction values. (top) matrix
difference $|A_{1}-A_{2}|$. Both top and bottom panels are averaged over no-
oscillation (n=40) and with-oscillation (n=62) training runs, thresholded at 1
kT. Polymer ends are positioned at the center of interaction matrices. (D)
Fraction of interactions which change by $>2$ kT to switch between $G1,G2$.
Violin plots show distribution over no-oscillation training pairs (n=40) and
with-oscillation pairs (n=62). Lines indicate minimum, mean, and maximum
values.

In Fig. 2, we showed that adaptable allosteric response can be created in
elastic networks by alternating training for two incompatible motions. This
training relied on the optimization of a global cost function under gradient
descent. However, recent examples of elastic network training aim to change
bulk elastic moduli with algorithms which use local information as input, and
modify the network in an irreversible fashiongoodrich2015principle ;
reid2018auxetic . When training for adaptability in mechanical allostery, we
implicitly assumed that the dynamics of training would allow for returns to
previously visited regions of design parameter space. It is not clear that we
can train for adaptability without the ability to move unrestricted through
design parameter space. Here, show that we can extend our oscillatory training
framework to the task of developing bulk elastic response even with
irreversible local update rules.

Our two target goals $G1$ and $G2$ now correspond to having Poisson ratios of
$>.75$ and $<-.75$ respectively (Fig. 3A). We will use the same notation as in
the previous section - $G1$ and $G2$ \- to refer to these goals. The Poisson
ratio $\nu$ of a material describes its bulk deformation response to a
uniaxial strain. If the applied strain is compressive, a negative $\nu$
indicates that a network will contract along the axis orthogonal to the
strain, while a positive $\nu$ indicates that a network will expand along the
orthogonal axis. Note that for isotropic materials, $\nu$ is constrained to be
within $[-1,1]$, but here we consider materials which may become anisotropic
as they are trained; see Supplementary Information for further detail.

Our algorithm for training elastic networks to perform $G1$ and $G2$ proceeds
by the irreversible removal of bonds based on local information. Our design
parameters are the presence or absence of a bond in the lattice, but we do not
allow bond additions. We initialize our training with disordered 2D mass-
spring networks of approximately 200 nodes. We simulate these lattices under
periodic boundary conditions. During training, we enforce a deformation on the
network, measure the strain in each bond, and then remove the bond which
experiences the most strain. When we train for positive $\nu$, we compress the
network along the y-axis and stretch it along the x-axis. Analogously, we
train for negative $\nu$ by compressing the network along the y-axis while
also compressing it along the x-axis. During oscillatory training, we
alternate which of these deformations is applied. See Supplementary
Information for further detail.

Despite the differences between the goals and algorithms considered here
compared to those utilized in the design of adaptable allostery, we find that
a similar picture of adaptable mechanical design under oscillatory training
emerges (Fig. 3B-D). Even when trained from the same initial disordered
network, training for $G1$ and $G2$ produce pairs of networks with many
differences in their bond removals (Fig. 3B(left)). In contrast, when we
oscillate which deformation is applied once every 20 bond removals, we find
that the difference between such network pairs is substantially lower (Fig.
3B(right)). This alternation comes at a cost; in an ensemble of networks
undergoing oscillatory training, approximately $50\%$ experience mechanical
failure before the training ends, due to the irreversible nature of bond
removals. Of those that survive the training, $12\%$ are able to rapidly
switch between $G1$ and $G2$, with an overall yield of $6\%$ (Fig. 3C). When
comparing an ensemble of networks trained with oscillation to an ensemble of
networks trained without, we find quantitative evidence that adaptability
increases with oscillatory training, at the cost of lower yield (Fig. 3D).

### Heterpolymer folding

Having demonstrated our method’s success in designing adaptable mechanical
networks, we turned to another paradigmatic class of tunable synthetic
materials.

Programmable self-assembly of single target structures has been explored in
many systems, ranging from colloids to proteins and DNA. Across these diverse
systems, a similar set of design parameters are tuned to target assembly of a
desired structure. Typically, these parameters include a matrix of binding
affinities between building blocks, in addition to global parameters like
temperature and concentrations. We refer to the matrix of binding affinities
as the affinity matrix.

In most approaches to self-assembly go1983theoretical ; pande2000heteropolymer
, the affinity matrix closely resembles the contact matrix of the building
blocks in the desired structure $\Gamma$. That is, particles in contact in
$\Gamma$ should typically have stronger binding affinities compared to
particles not in contact in $\Gamma$, thereby energetically stabilizing the
structure relative to other configurations of the same particles.

As a result, design parameters optimal for assembling a structure $\Gamma_{a}$
would not be good at assembling an unrelated structure $\Gamma_{b}$ with high
yield. This makes adaptability in self-assembly seem difficult from the
outset. The stochastic nature of self-assembly provides an additional
complication compared to elastic networks.

To test whether we can design a self-assembling system to be adaptable, we
built a simulation of 2-d heteropolymer folding using the HOOMD-blue
softwareanderson2020hoomd . Specifically, we consider a polymer of 13
monomers, each of which is bonded to the next with harmonic springs. A
harmonic bending energy is present to stabilize the fully-extended polymer
state with a persistence length of 5.5. Each monomer interacts with all non-
neighbor monomers through an attractive Morse potential with a tunable
affinity. The affinities have a maximal value of 6 kT and a minimal value of 0
kT. As such, the design space has dimension 66, equal to the number of lower-
triangular entries in a $13\times 13$ affinity matrix without including the
first off-diagonal; see Supplementary Information for further simulation
details.

Our task is to select affinity matrices which can readily switch between two
distinct, mutually exclusive design goals (Fig. 4A). Our first goal, $G1$, is
to produce polymers that fold from a fully-extended initial condition into a
spiral with a winding number exceeding that of 1.0 around the monomer at the
head of the polymer. As self-assembly is stochastic, we further specify that
this must occur with greater than 70% probability within $500$ units of
simulation time. $G2$ is defined analogously, with the difference being that
the target structure is now an anti-spiral, which has a winding number
exceeding that of 1.0 measured relative to the monomer at the tail of the
polymer. Note that use the same notation as in previous sections - $G1$ and
$G2$ \- to refer to these goals.

We optimize the yield of a given target structure over the 66 design
parameters using the Covariance Matrix Adaptation-Evolutionary Strategy (CMA-
ES) hansen1996adapting ; hansen2006cma  that simulates an evolving population
of design parameters. The loss function for our implementation of CMA-ES is
the negative of the yield, with a floor set by the minimum $70\%$ yield
required for successfully achieving either $G1$ or $G2$; see Supplementary
Information for further parameter choice details.

We perform optimization with two training protocols: 1. “no-oscillation”
training, where $G1$ and $G2$ are optimized individually, and 2. “with-
oscillation” training, where we switch between $G1$ and $G2$ with a period of
$5$ training steps. When we successfully perform optimization in with-
oscillation training, we see that the maximal yield of each goal increases in
an alternating fashion with each passing training period (Fig. 4B). This
suggests that oscillating training is converging to affinity matrix solutions
which can easily switch between $G1$ and $G2$. Through this procedure, we
collect a set of affinity matrix pairs $A_{1},A_{2}$. Additionally, we verify
that the adaptability of our pairs $A_{1},A_{2}$ do not come at the expense of
significant performance degradation on the individual goals they are trained
for (Supplemental Fig. 3). We collect an analogous set of pairs for no-
oscillation training simply by running converged optimizations from the same
initial conditions.

To characterize the distribution of successful affinity matrix pairs, we
computed the average difference between a matrix that achieved $G1$ and its
corresponding $G2$ partner (Fig. 4C, top row), focusing only on entries that
changed substantially (i.e., by more than $1$ kT). The resulting average
difference matrix from the no-oscillation training shows many more changed
entries than the matrices from with-oscillation training. This visually
suggests that oscillatory training identifies more adaptable regions of design
space, where the affinity matrices which achieve $G1$ are closer to those
which achieve $G2$. The average affinity matrix for each goal supports the
same conclusion (Fig. 4C, bottom row). Note that we have shifted the monomer
numbering when plotting affinity matrices, for ease of visual comparison.

To quantitatively confirm these visual conclusions, we compute the fraction of
matrix entries that change by more than $2$ kT between each $A_{1},A_{2}$
affinity matrix pair (Fig. 4D). As expected, the distribution of this metric
across all such pairs is substantially higher for no-oscillation training than
for with-oscillation training.

#### Physical interpretation

Figure 5: Adaptability in polymer folding relies on localizing interaction
changes to nucleation barriers. A. When trained without oscillations, energy
landscapes (cartoons) targeting $G1$ (green) and $G2$ (orange) have deep
energy minima at their on-target $G1$ and $G2$ (black open circle) but high
energy at off-target $G2$ and $G1$ (black solid circle) respectively. With
oscillatory training, discrimination is now kinetic; both on- and off-target
energies remain low, but the landscape develop kinetic barriers to folding
(white cross black squares). Kinetic barriers enable adaptability since
folding can changed by changing a small number of contacts in the nucleation
seed. B. Off-target energy distributions for no-oscillation (left) and with-
oscillation (right) training (data from 40 and 62 simulation runs
respectively); energy is relative to the mean of the no-oscillation on-target
distribution. The off-target distribution is lower in energy than in with-
oscillation training. Violin plot lines indicate minima, means, and maxima. C.
As in Fig. 4C, interaction matrix differences between $G1$ and $G2$, for no-
oscillation (left) and with-oscillation (right) training. Barrier contacts are
overlaid as white cross black squares; oscillatory training localizes
interaction matrix changes to the barrier contacts. Monomers are colored
according to position.

The adaptability of self-assembly found here is surprising at first glance.
The average adaptable pair of affinity matrices $A_{1},A_{2}$ resemble each
other for the majority of elements (Fig. 4C, right), yet fold into
incompatible configurations with high yield.

To understand the physical design principles underlying such adaptability, we
estimated aspects of the folding energy landscape for affinity matrices $A$
found through oscillatory and non-oscillatory training; we computed the
energies of folded configurations with different winding number (see
Supplementary Information).We find that on-target structures are similarly
stabilized by both oscillatory and non-oscillatory training as suggested by
the cartoon in Fig. 5A (open circles).

However, the two training protocols differ in how they treat off-target
structures. With non-oscillatory training, the off-target structures are
relatively high in energy (solid circle in Fig. 5A,B) since training is only
ever shown the on-target structure. Consequently, the affinity matrix requires
extensive changes to assemble the previously-off-target structure (Fig. 5C).

In contrast, with oscillatory training, off-target structures remain low
energy states even with the affinity matrix that target a radically distinct
on-target structure. Relative to the no-oscillation off-target distribution,
the with-oscillation off-target distribution is lower by $\sim 10$ kT (Fig.
5B).

Despite such energetic stabilization of off-target structures, with-
oscillation training results in on-target folding by exploiting kinetics.
Folding is controlled by a nucleation barrier that is higher for the off-
target structure than for the on-target structure (cartoon Fig.5A; right).

Using the estimated energy landscapes, we can identify “barrier contacts” that
need to form in a partially folded nucleation seed before subsequent downhill
folding to completion. Oscillatory training localizes the few changed
affinities to those involved in forming barrier contacts (black in Fig. 5C).

Thus, our time-varying algorithm points at a physical principle for adaptive
self-assembly of independent validity. Kinetic yield is controlled by
partially folded early intermediate structures that correspond to nucleation
barriers. These barriers can be lowered in energy or conversely destabilized
by relatively few changes to the affinity matrix, resulting in the spiral or
anti-spiral with high selectivity. Similar principles might apply more broadly
to proteins and ribozymes where partly folded configurations, en route to
fully-folded configurations can be destabilized; indeed, such mechanisms might
operate in experimentally characterized adaptable proteins and ribozymes where
a single mutation can switch the polymer between distinct structures and thus
function.

## Discussion

Biological materials differ from man-made materials in not just their physics
and composition but also in the history of their development. In fact, the way
biological materials are arrived at, through a process of incremental
evolution in a sequence of historic environments, is critical in understanding
why they function differently from man-made materials. While man-made
materials can sometimes rival or even exceed specific functionalities of
natural systems, these synthetic systems are lagging precisely in meta-
properties like adaptability, robustness and ability to acquire new functions
on the fly.

Here, we have shown how one such property, adaptability, can arise without any
direct targeting or optimization. Instead, we find adaptable materials by
applying a time-varying sequence of selection pressures during design
optimization.

This adaptability comes through the spontaneous formation of identifiable
physical units, despite the fact that our goals and systems were not
explicitly modular in form. In contrast, prior workkashtan2005spontaneous ;
hemery2015evolution  used explicit modularly varying goals to show a potential
origin of modularity and adaptability in biological systems. Thus our work can
be seen as building on those ideas to discover physical principles specific to
the physics of systems studied (e.g., nucleation for polymer folding) that
allow for adaptability.

Our proposed method has potentially wide applicability, in that it functions
as a wrapper around pre-existing design programs, and hence can be applied
without in-depth knowledge of a system.

However, our method can also help reveal new system-specific physical insights
that can then be exploited without further need for our method. For example,
in many of the current platforms for self-assembly, the yield is frequently
governed by kinetics rather than equilibrium free energiesjacobs2015rational ;
hensley2022self ; bitran2020cotranslational . Our simulations of heteropolymer
self-assembly revealed a broadly relevant design principle for such systems –
nucleation barriers in energy landscapes can be leveraged to create
adaptability in self-assembly. Similarly, in the context of elastic networks,
we identified that coherent motions which link two allosteric sites can be
easily diverted in order to achieve incompatible goals. These insight can now
be used to guide design without need for blind numeric optimization, both in
synthetic systems like colloids, DNA, and foams, but also in natural systems
like proteinsbitran2020cotranslational .

One key condition for our method is already clear from the pilot examples in
this study. In order for our adaptability framework to succeed, the goals
under consideration must have “neutral variation”Gruner1996-ll ; Gruner1996-dj
; Schultes2000-mg ; there must exist changes in design parameters that have no
cost in terms of the current target functionality but that help adapt to new
functionality. The systems studied here have such degeneracy; for example,
many elastic networks with different bond stiffnesses showed the same desired
allosteric response. In fact, degeneracy is generically expected whenever
systems are disordered, with the number of design parameters often being
extensive in the size of the system. In biological examples too, such
genotype-to-phenotype maps are often redundant when the space of genotypes (or
design parameters) is larger dimensional than the space of phenotypes
(exhibted properties). We do not expect as much success in systems such as
self-assembled crystals with only 1 or 2 species or regular lattices of
elastic elements.

Acknowledgments. The authors thank Varda Hagh, Heinrich Jaeger, Ilya Nemenman,
Rama Ranganathan, Riccardo Ravasio, and Kabir Husain for discussions. This
work was primarily supported by the University of Chicago Materials Research
Science and Engineering Center, which is funded by the National Science
Foundation under award number DMR-2011854. Arvind Murugan acknowledges support
from the Simons Foundation. M.L.G. acknowledges support from NSF Grant DMR-
2215605. Ayanna Matthews acknowledges support the University of Chicago
Biophysics Training Grant.

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