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\DOI{10.5802/crchim.389}
\datereceived{2024-11-13}
\daterevised{2025-02-14}
\dateaccepted{2025-02-26}
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\dateposted{2025-04-08}
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\begin{noXML}

\CDRsetmeta{articletype}{review}

\title{Shortest Path Map correlation-based tool for capturing
functionally relevant allosteric networks and its application in enzyme
design}

\alttitle{Outil Shortest Path Map \`{a} base de corr\'{e}lations pour
capturer les r\'{e}seaux allost\'{e}riques fonctionnellement pertinents
et son application dans la conception d'enzymes}

\author{\firstname{Esther Pruna} \lastname{Cortada}\CDRorcid{0009-0004-7199-2315}}
\address{Institut de Qu\'{i}mica Computacional i Cat\`{a}lisi and
Departament de Qu\'{i}mica, Universitat de Girona, Parc R+i Univ.
Girona, Ed. Monturiol, c/ Emili Grahit 91, 17003 Girona, Spain}

\author{\firstname{S\'{i}lvia} \lastname{Osuna}\CDRorcid{0000-0003-3657-6469}\IsCorresp}
\addressSameAs{1}{Institut de Qu\'{i}mica Computacional i Cat\`{a}lisi
and Departament de Qu\'{i}mica, Universitat de Girona, Parc R+i Univ.
Girona, Ed. Monturiol, c/ Emili Grahit 91, 17003 Girona, Spain}
\address{ICREA, Pg. Llu\'{i}s Companys 23, 08010 Barcelona, Spain}
\email[S. Osuna]{silvia.osuna@udg.edu}

\keywords{\kwd{Molecular dynamics simulations}\kwd{Correlation-based
tools}\kwd{Enzyme design}\kwd{Shortest Path Map}\kwd{Allostery}} 

\altkeywords{\kwd{Simulations de dynamique mol\'{e}culaire}\kwd{Outils \`{a} base de corr\'{e}lations}\kwd{Conception d'enzymes}\kwd{Shortest Path Map}\kwd{Allost\'{e}rie}}

\begin{abstract}
The impact of distal mutations on enzyme design is analogous to the
allosteric regulation effect observed in effector binding or substrate
transport within heterocomplexes. Building on this analogy, we
employed molecular dynamics simulations to estimate the conformational
landscape of enzymes and developed the Shortest Path Map (SPM) tool.
This tool identifies  key conformationally relevant positions that
regulate enzymatic function. In this review, we highlight various
applications of the SPM method in its use as a tool for investigating
the effects of distal mutations on the conformational landscape,
characterizing allosteric regulation, and rationally designing enzymes.
A brief description of the method is provided together with several
examples reported over the years (from us and other labs) in which the
SPM method has been successfully applied.
{\vspace*{-2pt}}
\end{abstract}

\begin{altabstract}
L'impact des mutations distales sur la conception des enzymes est
analogue \`{a} l'effet de r\'{e}gulation allost\'{e}rique observ\'{e}
dans l'affinit\'{e} des effecteurs ou le transport des substrats au
sein des h\'{e}t\'{e}rocomplexes. Sur la base de cette analogie, nous
avons utilis\'{e} des simulations de dynamique mol\'{e}culaire pour
estimer le paysage conformationnel des enzymes et d\'{e}velopp\'{e}
l'outil Shortest Path Map (SPM). Cet outil identifie les positions
conformationnelles cl\'{e}s qui r\'{e}gulent la fonction enzymatique.
Dans cette revue, nous mettons en lumi\`{e}re diverses applications de
la m\'{e}thode SPM en tant qu'outil d'\'{e}tude des effets des
mutations distales sur le paysage conformationnel, de
caract\'{e}risation de la r\'{e}gulation allost\'{e}rique et de
conception rationnelle d'enzymes. Une br\`{e}ve description de la
m\'{e}thode est fournie, ainsi que plusieurs exemples rapport\'{e}s au
fil des ans (par nous et d'autres laboratoires) dans lesquels la
m\'{e}thode SPM a \'{e}t\'{e} appliqu\'{e}e avec succ\`{e}s.
\end{altabstract}

\maketitle

\twocolumngrid

\end{noXML}

\section{Introduction} \label{sec1}

{\vspace*{-2pt}}

Enzyme design seeks to develop biocatalysts with improved ``\`{a} la
carte'' characteristics by altering the amino acid sequences of natural
enzymes or generating new sequences and tertiary structures. The
interest in enzyme design and engineering stems from the unique
potential benefits of these catalysts, such as their ability to work
efficiently under mild biological conditions while achieving
exceptional efficiency, selectivity, and specificity. Enzyme design is
intellectually stimulating, serving as a stringent test of our
understanding of enzyme stability, folding, evolution, and catalysis.

The process of enzyme design, whether starting from a natural or
computationally generated scaffold, involves selecting specific
residues for \mbox{mutation}, creating new variants, and applying screening
methods to evaluate the desired properties~\cite{1}. There are two
primary approaches that can be effectively combined: rational
design~\cite{2,3,4} and directed evolution (DE)~\cite{5,6,7}. Rational
design focuses on predetermined key residues, identified through the
analysis of multiple sequence alignments (MSAs), structural evaluation of
active sites, substrate-binding tunnels, reaction mechanisms, and key
intermediates formed during the reaction, using techniques like quantum
mechanics, molecular mechanics, molecular dynamics (MD), and Monte Carlo
simulations~\cite{3,8}. Rational methods typically target as mutation
hotspots active site residues~\cite{9,10} or those involved in the
bottleneck regions of tunnels, using tools like CAVER, AQUA-DUCT, and
HotSpot Wizard~\cite{8,11,12}. Inspired by the deep-learning (DL) AF2
approach~\cite{13,14,15}, recognized with the 2024 Noble Prize
in Chemistry, many DL techniques have been recently developed for
generating new protein sequences and scaffolds, which then are further
refined using sequence design methods such as
ProteinMPNN~\cite{16,17,18,19,20,21,22}. All these different designs
generated by means of traditional methods or DL can also be further
improved with DE. Recognized with the 2018 Nobel Prize in
Chemistry, DE traditionally utilized random mutagenesis but has advanced
to integrate bioinformatics~\cite{23,24,25,26,27,28}, rational
methods~\cite{29,30}, sequence analysis~\cite{31,32}, NMR
data~\cite{33}, and high-throughput screening~\cite{34}, often guided
by machine-learning models~\cite{35,36}. The DE approach is particularly powerful
for improving low-activity computational enzyme designs~\cite{37,38,39}
and enhancing side activities~\cite{40,41}. Numerous lab-engineered
enzymes have been developed over the years, contributing to drug
production, biotherapeutics, and fragrance production~\cite{42}.

One key advantage of DE is its ability to introduce mutations across
the entire protein unlike traditional rational design strategies,
which often focus on the active site or substrate tunnels for helping
substrate binding/product release and/or altering the water
content~\cite{8,43}. The DE studies have shown significant activity
increases through mutations distant from the active site, which are
difficult to predict computationally~\cite{6,44,45,46}. This has been
observed in various enzyme families, where in many cases mutations
occur around 15~\AA{} from the active site~\cite{44}. Intriguingly, the
impact of mutations on the enzyme turnover ($k_{\mathrm{cat}}$) is
not directly correlated to their proximity to the active site, which
contrasts with the more deterministic role of active site mutations in
specificity~\cite{6}. Such regulation of the enzyme catalytic activity
by means of distal mutations suggests that allostery (i.e., regulation
of catalytic activity by effector and/or protein binding) plays a key
role in many proteins~\cite{47}. Molecular dynamics  simulations
have explained how such mutations influence enzyme conformational
dynamics and enhance catalytic activity by altering the network of
non-covalent interactions~\cite{46,48}, thus regulating the flexibility
of dynamical elements such as loops or lids gating the active site
access~\cite{40,49,50}. Although computational models can rationalize
these changes, predicting which distal mutations will affect activity
remains challenging~\cite{46,48,51}. The insights gained from DE about
the role of distal mutations regulating enzymatic function could help
develop computational approaches for designing efficient enzymes with
natural-like functionality~\cite{44}.

The influence of distal mutations in enzyme design parallels the
allosteric regulation seen in effector binding or substrate transport
within heterocomplexes. These mutations can shift the conformational
landscape of the enzyme, favoring a subset of conformations that
enhance catalysis. Recognizing this similarity, we explored the
application of MD simulations for estimating the enzymes'
conformational landscape and extracting key conformationally relevant
positions for their use in enzyme design~\cite{2,48}. We developed the
Shortest Path Map (SPM) tool~\cite{52}, which constructs a graph based
on mean distances and correlation values between residues as
determined from MD simulations. Unlike previous
allosteric studies that identified regions of importance~\cite{53}, the
SPM method identifies specific residue pairs that significantly
influence enzyme dynamics (Figure~\ref{fig1}). This precise
identification of key residues is valuable in enzyme design, enabling
the creation of targeted libraries for mutagenesis.

\begin{figure*}
{\vspace*{2pt}}
\includegraphics{fig01}
{\vspace*{2pt}}
\caption{\label{fig1}Computational enzyme design based on inducing a
population shift on the enzyme conformational landscape towards active
states by using the SPM method. Left: The wild-type enzyme
conformational landscape features multiple stable states, which are
predominantly catalytically inactive (labeled  Conf~1, Conf~2,
Conf~3, and Conf~4). This represents the enzyme conformational
heterogeneity, but the enzyme presents poor catalytic efficiency as
active states are poorly populated. Center: By computing the distance
and the correlation matrices from the multiple replica MD simulations,
the first complex graph network is generated, which is further simplified
to obtain the SPM network. SPM identifies potential
mutational hotspots that can influence the global conformational
landscape and induce a shift in the population distribution towards
catalytically active states. Right: By introducing targeted mutations
at both active and distal sites, the conformational landscape is
reshaped to favor catalytically active states. This approach has a wide
range of applications, including directed evolution mutation
identification (Section~\ref{ssec21}), elucidation and optimization of
allosteric pathways (Section~\ref{ssec22}), and finally, rational
enzyme design  (Section~\ref{ssec23}).} 
{\vspace*{4pt}}
\end{figure*}

\section{SPM tool}\label{sec2}

The SPM builds a graph based on mean distances and correlation values
computed from MD simulations, following a strategy similar to the
protocol used by Sethi et al.~\cite{53}. For each protein residue, a
node is created centered on the carbon alpha atom. The next step is to
establish edges between node pairs. An edge is formed between pairs of
nodes whose C$_{\upalpha}$ atoms remain within 6~\AA{} of each other
throughout the MD simulation. The edge distance is then derived from
the calculated correlation values, which represent the information
transfer along that edge. Residue pairs with higher correlation values
(closer to 1 or ${-}$1) have shorter edge distances while less
correlated pairs (values closer to 0) have longer edge distances. This
approach results in a graph with nodes and edges reflecting residue
proximity and correlation, which is then simplified as illustrated in
Figure~\ref{fig1}.\looseness=-1

Unlike previous allosteric studies that focus on identifying
communities within the first complex graph generated, we compute the
shortest path lengths. The SPM identifies which edges of the graph are the
shortest (i.e., most correlated) and most frequently used when passing
through all the protein residues. After normalizing all edges, only
those with the highest contribution are depicted and visualized in the
3D structure. The key advantage of the SPM is that it directly pinpoints
critical residues rather than broad regions, making it especially
valuable for enzyme design by enabling the construction of small
libraries of hotspot positions.

The SPM has proven effective in identifying mutation spots, either within
the active site or at distal locations targeted by DE. It has also been
applied for elucidating the allosteric regulation existing in
multimeric protein structures and more recently for enzyme design. The
following sections discuss several cases where SPM has been applied:
(1) identification of DE mutation hotspots, (2) rationalization of allosterically
relevant residues, and (3) for enzyme design.

\subsection{SPM captures DE mutations}\label{ssec21}

The DE approach has become an invaluable strategy in enzyme design engineering,
enabling the discovery of enzymes with improved or novel
functionalities through iterative rounds of mutation and
selection~\cite{5,6,7}. The key advantage of DE is its ability to
identify beneficial mutations across the entire protein, which are often
difficult to predict computationally, but it lacks mechanistic insights
into how the identified mutations affect enzyme dynamics and
conformational changes. Computational models can rationalize DE
mutations, but the challenge persists in the prediction of these key
mutations. 

The SPM's predictive capabilities have been validated in several key case
studies, where it successfully identified mutations that mirrored those
obtained through DE. For instance, in the case of mechanistically
complex \textbf{retro-aldolase (RA) enzymes}, DE introduced multiple
mutations scattered throughout the enzyme, which drastically increased
its catalytic activity towards the abiological aldol substrate
methodol~\cite{23,45,48}. The SPM tool could predict most of the mutational
hotspots observed in the experimentally evolved enzymes. Specifically,
the SPM identified seven of the thirteen mutations introduced by DE while
four others were located at adjacent positions and only two were more
than six positions away in the sequence (Figure~\ref{fig2}a). Of note
is that designed RAs are $(\upbeta\upalpha)_{8}$-barrel enzymes, which
is a fold shared by many enzymes suggesting that the application of the SPM
tool might be quite broad. A similar result was observed in the complex
homodimeric \textbf{monoamine oxidase from \textit{Aspergillus
niger} (MAO-N)}~\cite{49,54,55}, where DE was employed to
increase catalytic activity towards chiral amines and broaden the
enzyme's substrate scope~\cite{56}. Five positions were introduced by
DE for obtaining the MAO-N D5 evolved variant~\cite{57}. The key DE
mutations include one active site mutation, one mutation at the
entrance tunnel, and three additional distant mutations up to
18--19~\AA{} away from the active site, all of them, despite being
computationally challenging to predict, identified by the SPM tool or
located at adjacent positions (Figure~\ref{fig2}d). The computational evaluation of the
conformational changes induced by these mutations by means of Markov
state models indicated major changes in beta-hairpin
conformation between wild-type and D5 variants. Additionally, the SPM
applied to D5 contains the DE positions for generating D9 and D11
variants~\cite{57}. This agreement between computational predictions
and experimental outcomes underscores the SPM's effectiveness in
identifying functionally relevant distal positions and highlights its
ability to predict key residues regulating enzyme activity by analyzing
the conformational dynamics of the starting enzyme~\cite{44}. 

\begin{figure*}
{\vspace*{-2pt}}
\includegraphics{fig02}
{\vspace*{-2pt}}
\caption{\label{fig2}Some examples of the application of the SPM
method for capturing DE mutations. (a)~SPM is applied to retro-aldolase
enzyme. Of the thirteen mutations introduced by DE, seven were directly
identified by SPM (green) with an additional four located at adjacent
positions (yellow). Only two mutations (purple) were positioned more
than six residues away in sequence. (b) Six distal mutations in $Pf$TrpB
were introduced by DE to generate a stand-alone $Pf$TrpB-0B2 variant. Of
these, five mutations were identified by SPM: two directly captured by
SPM (green) and three adjacent to SPM-identified positions
(orange). The only distal mutation not captured by SPM is shown in
purple. (c) The SPM in 1A53-2 (left) and 1A53-2.5 (right) depicts the
interaction network centered on the transition state  and extending to
the closed state of the protein. The two mutations found by DE are
directly involved in the SPM. (d) Right: The five mutations in MAON-WT
identified by DE to generate MAON-D5 evolved variant are shown as
spheres. The key DE mutations include one active site mutation
(purple), one mutation at the entrance tunnel (yellow), and three
additional distant mutations (green). Left: The DE mutations on MAON-D5
towards MAON-D9 variant are also captured by SPM and are shown in dark
green while previous mutations are shown in brown. The beta-hairpin
structure is highlighted in teal, and the entrance loop is represented
in violet in both images. The FAD cofactors are depicted as sticks in
dark violet. (e) The SPM pathway of the CYP450 enzyme included the
previously identified DE mutations V78 and A330 (green). SPM also
depicts the long-distance connection between beta-sheet 1 (purple) and
beta-sheet 4 (light violet). The key mutation Y51I, identified by SPM as
an adjacent position, is shown in yellow.}
{\vspace*{-2pt}}
\end{figure*}

The SPM tool was also employed by the Mulholland Laboratory to analyze
shifts in dynamical networks within the transition-state (TS) ensemble
along the DE of a computationally designed \textbf{Kemp eliminase}, which
catalyzes a proton abstraction from carbon by a base~\cite{58}. An SPM
analysis of the closed states uncovered a dynamical network that
connected the active site to more distal regions of the protein, which
indicates fine-tuning of its dynamics by remote mutations
(Figure~\ref{fig2}c). Mutations
such as Q211G and L184F, introduced during DE, were also identified by
SPM as key contributors to this network, facilitating the global
response of the enzyme scaffold to the TS. Mutation Q211G increased flexibility,
fine-tuning the dynamic response while L184F enhanced packing by
tightening the interactions between neighboring solvent-exposed loops,
further stabilizing the TS. These mutations were essential in reducing
the structural fluctuations in the TS ensemble, thereby improving
catalytic preorganization. 

Another striking example of SPM's utility to identify the mutations
introduced by DE comes from work on the heterocomplex
\textbf{tryptophan synthase from \textit{Pyrococcus furiosus}
(\textit{Pf}TrpS)}. 
The TrpB complex, a subunit of the
tryptophan synthase complex (TrpS), was subjected to DE to create an
efficient stand-alone enzyme variant independent of the alpha subunit
(TrpA) with improved catalytic activity~\cite{59,60}. The evolved
variant, \textit{Pf}TrpB-0B2, contained six distal mutations that
restored the enzyme's ability to access essential conformational states
in the absence of its TrpA protein binding 
partner~\cite{61,62} (Figure~\ref{fig2}b). Two of
the distal mutations, P12L and E17G,  introduced by DE are located near the TrpA--TrpB
interface. Another mutation, F247S, is found in a known
tunnel while T292S is situated in one of the loops close to the
essential COMM domain that covers the active site where the pyridoxal
phosphate cofactor is located. Additionally, two more mutations, I68V
and T321A, are present in more remote regions of the enzyme. Of the six
mutations in \textit{Pf}TrpB-0B2, the first two were directly predicted
by SPM and three were making persistent non-covalent interactions with
SPM-identified positions. Mutation T292S, which significantly enhanced enzyme
activity, modulated COMM domain closure via interactions with D300,
supporting SPM predictions. The only residue not captured by the SPM is
T321, which is considered to play a minor role in COMM domain
conformational dynamics. The comparison of the free energy landscapes
between the wild type and the evolved \textit{Pf}TrpB-0B2 indicated
that these distal mutations recovered the conformational heterogeneity
observed for the wild type when forming complexes with TrpA~\cite{61}. This
study highlights the potential of the SPM in identifying key residues for
improving enzyme function, offering a powerful tool for targeting
allosterically regulated systems. 

The SPM was also applied to study the epistatic effects and conformational
dynamics in the stepwise evolution of a \textbf{cytochrome P450-BM3
monooxygenase} engineered for regioselective and stereoselective
hydroxylation of a steroid~\cite{63}. The previously identified DE
mutations V78 and A330 were \mbox{successfully} included in the SPM pathway of
the parent enzyme (Figure~\ref{fig2}e). Additionally, combinatorial saturation mutagenesis
revealed that the mutant R47I/T49I/Y51I/F87A exhibited improved
activity compared to the wild type, achieving 94\%
2${\upbeta}$-selectivity and 67\% substrate conversion. Computational
analysis showed that while mutations R47I and T49I alone did not affect
selectivity, the Y51I mutation shifted substrate orientation,
increasing \mbox{2${\upbeta}$-selectivity.} The SPM
successfully identified the Y51 position providing the key residue
responsible for both activity and selectivity. Furthermore, the SPM
pathway detected a long-distance communication network between
${\upbeta}$4 and ${\upbeta}$1, involving positions R47, T49, and Y51. 

These selected examples show that the SPM method is a valuable
computational tool for capturing the conformational effects of
mutations introduced by DE. By pinpointing key residues and pathways
that regulate conformational transitions, the SPM can be, in principle,
used to enhance the efficiency of enzyme engineering efforts by
\mbox{predicting} conformationally relevant positions not strictly
located at the active site. The successful application of the  SPM in
enzymes like RA, monoamine oxidase, tryptophan synthase, Kemp
eliminase, and cytochrome P450 illustrates its potential to complement
and rationalize DE approaches, accelerating the development of enzymes
with optimized functionalities through a deeper understanding of their
conformational landscapes.

\subsection{SPM identifies allosteric pathways}\label{ssec22}

Allosteric regulation is a fundamental process in which distant
residues within a protein communicate to regulate its activity. This
long-range communication allows proteins to respond to various signals,
adjusting their function accordingly. Understanding the specific
pathways through which allosteric signals travel is crucial to enzyme
design and for understanding how mutations alter protein function. The SPM
has proven effective in mapping allosteric pathways in key proteins,
offering insights into how mutations can alter their regulation and
function. For instance, the SPM was applied to 5~$\upmu$s atomistic MD
simulations of the \textbf{K-Ras4B protein} embedded in a phospholipid
membrane to investigate its allosteric regulation~\cite{64}. Allosteric
regulation in K-Ras, particularly the oncogenic K-Ras4B, is key to its
function in cellular signaling. K-Ras switches between an active
GTP-bound state and an inactive GDP-bound state with allosteric
changes in key regions of the G domain (switches I and II), enabling
effector recruitment and signal transduction. Mutations, especially in
Gly12 and Gly13, can disrupt this regulation, locking K-Ras in an
active state and \mbox{driving} cancer 
progression~\cite{64}. The SPM identified a
prominent allosteric pathway starting at the hypervariable region
(HVR), a near-membrane region that might be receiving allosteric
messages and transferring them to the major hub, located on helix
{${\upalpha}$}5, loop {${\upalpha}$}3--{${\upbeta}$}5, and the
N-terminus of sheet {${\upbeta}$}5 (Figure~\ref{fig3}a). Branching out to key areas in the G
domain, the allosteric route extends to regions crucial to K-Ras
function, including the P-loop, Thr58, and Ala59 on switch II and Thr35
on switch I. The analysis of the allosteric communication existing in
K-Ras4B was also assessed by means of other measures: distance
fluctuation analysis~\cite{65,66}, anisotropic thermal
diffusion~\cite{67}, and dynamical non-equilibrium
simulations~\cite{68}, which provided complementary information on how
allostery is transmitted when hydrolyzing GTP~\cite{64}. The SPM therefore
provided useful insights into K-Ras allostery that help in identifying
potential allosteric sites for therapeutic targeting, advancing drug
design efforts for this\break protein. 

\begin{figure*}
\includegraphics{fig03}
\caption{\label{fig3}Some examples of the application of the SPM method
for identifying allosteric networks. (a)~K-Ras4B: SPM reveals the
allosteric network connecting the membrane-embedded HVR region (dark
red) to the central hub (violet), which extends to switch II (orange)
and further connects to switch I (green). Notably, the critical
mutations Gly12 and Gly13 in the P-loop (blue) are also captured by
SPM. (b) Hsp60: The allosteric network links the equatorial domain
(wheat), intermediate domain (white), and apical domain (dark red).
Introducing the V72I mutation disrupts this network, disrupting the
communication between the equatorial domain and the active site,
represented by residue A397 in dark orange. (c) IGPS: SPM analysis
shows the progression of IGPS from the open HisH--HisF interface to an
allosterically active state, where multiple pathways connect HisF and
HisH active sites, facilitating the formation of the hV51 oxyanion
hole.}
{\vspace*{-1pt}}
\end{figure*}

In a recent study, the allosteric regulation mechanism of the human
mitochondrial heat shock protein (\textbf{Hsp60}), which is responsible
for controlling proteostasis, is elucidated using the SPM
tool~\cite{69}. This study examines the effect of V72I point mutation,
linked to hereditary spastic paraplegia SPG13, which is characterized
by progressive weakness and spasticity of the lower limbs. An SPM analysis
was performed on MD simulations of both the wild-type (M WT) and mutant
(M V72I) Hsp60 monomers (Figure~\ref{fig3}b). The results indicate that the extensive
interdomain allosteric communication present in the wild-type monomer
is entirely disrupted in the V72I mutant, leading to decreased
efficiency in communication across the intermediate domain, which
contains the critical catalytic residue Asp 397. By mapping the
allosteric network in the Hsp60 monomer, this work offers valuable
insights into the pathogenic effects of the V72I mutation,
demonstrating how specific distal mutations can alter protein function
and contribute to disease and highlighting the potential applicability
of SPM in rationalizing allosterically regulated mechanisms. 

Imidazole glycerol phosphate synthase (IGPS) enzyme has been a model
for studying allosteric regulation~\cite{70,71,72}. The enzyme IGPS consists of
two \mbox{subunits}, HisH and HisF, with HisH catalyzing glutamine hydrolysis
and HisF delivering ammonia through an internal tunnel. Bound
30~\AA{} from the HisH active site, PRFAR acts as an allosteric effector,
enhancing glutaminase activity by 4500-fold~\cite{73}. In this case, a
time-evolution Shortest Path Map (te-SPM) analysis is applied to MD
simulations, specifically by examining accelerated MD  simulations
in 600~ns time intervals, uncovering the progression of
allosteric activation~\cite{74} (Figure~\ref{fig3}c). Initially, correlated motions are
limited to the HisH subunit and the HisF--HisH interface. However, as
the te-SPM analysis progresses, it reveals concerted motions that are
activated upon productive closure of the HisF--HisH interface, extending
throughout the entire HisF subunit, the interdomain region, and the
HisH active site, leading to the formation of the hV51 oxyanion hole.
This dynamic process aligns with ${\upmu}$s--ms motions observed by NMR
and suggests that IGPS allosteric activation follows a ``violin model''
of dynamics-based allostery, similar to what is found in some protein
kinases~\cite{75}. The truncated SPM approach for the time-evolution
analysis can be also used to decipher the molecular basis of allosteric
mechanisms in related allosterically regulated enzymes. 

All the provided examples support the idea that the SPM tool is a
useful strategy for uncovering the intricate mechanisms of allosteric
regulation in proteins by identifying critical pathways of residue
communication. So far, the SPM has offered deep insights into how proteins
modulate their functions and how mutations can disrupt this regulation.
Therefore, the SPM serves as a promising tool for rational design of
targeted therapeutics and enzyme engineering.

\subsection{Conformationally driven enzyme design with
SPM}\label{ssec23}

The application of the SPM in different unrelated enzymes demonstrates that
analyzing an enzyme conformational ensemble can pinpoint key positions
that significantly impact its conformational dynamics. Notably, the SPM
identifies positions throughout the enzyme, not just in the active
site, allowing for the prediction of distal mutations. This suggests a
strategy for conformationally driven enzyme design involving the
reconstruction of the enzyme conformational landscape, detection of
critical positions using the SPM, and identification of the specific amino
acid substitution for each mutation through experimental or
computational methods.

We engineered a \textbf{stand-alone functional TrpB} by combining the SPM
with ancestral sequence reconstruction (ASR)~\cite{76}. Interestingly,
ASR revealed that the last bacterial common ancestor (LBCA) is
allosterically inhibited by TrpA, thus presenting high stand-alone
activity~\cite{77,78}. Over the course of evolution, this inhibition
shifted towards allosteric activation; specifically ANC3 TrpS is the
first ancestral enzyme displaying allosteric activation by TrpA. By
coupling SPM and sequence conservation analysis between LBCA and ANC3
TrpB, six key positions were rationally identified, leading to the SPM6
variant, which achieved a sevenfold increase in catalytic
activity~\cite{76} (Figure~\ref{fig4}a). This improvement of activity obtained by the
experimental testing of a few ({\textless}5) variants was similar to that
obtained after multiple rounds of DE in \textit{Pf}TrpB that required
the generation and screening of more than 3000 variants~\cite{59}. Two
additional variants were developed that further validate the approach:
SPM3, based on ANC2 TrpB, involved three mutations;  SPM8, derived
from an additional SPM analysis of ANC3 TrpS instead of TrpB, added two
allosterically relevant positions~\cite{76}. While SPM8 achieved similar
catalytic improvement to SPM6, SPM3 showed a smaller enhancement,
which was expected from the reduced number of mutations. An additional
comparison of SPM with MSA revealed the
complementarity of the two methods, as they identified distinct
mutation sites with only one shared residue mutation between
them~\cite{76}. 

\begin{figure*}
{\vspace*{4pt}}
\includegraphics{fig04}
{\vspace*{4pt}}
\caption{\label{fig4}Some examples of the application of the
SPM method for rational enzyme design. (a) Designing stand-alone TrpB
variants based on the ancestral ANC3-TrpB scaffold: SPM identifies a
set of conformationally relevant positions in LBCA-TrpB (violet), which
after sequence analysis conservation with ANC3-TrpB residues led to the
SPM6 TrpB design exhibiting a sevenfold increase in catalytic activity
with respect to ANC3 WT. (b) Designing stand-alone TrpA enzymes: the
comparison of the SPM graphs computed considering IGP-bound
catalytically activated states of the stand-alone ZmBX1 (top) and
ZmTrpA-L6BX1 (bottom) allowed the generation of the ZmTrpA-SPM4-L6BX1
variant. The selected positions mutated are shown in purple for ZmBX1
and in yellow for ZmTrpA-L6BX1. ZmTrpA-SPM4-L6BX1 presents a 178-fold
higher stand-alone catalytic efficiency compared to ZmTrpA. (c)
Designing efficient esterases from hydroxynitrile lyases: the
introduction of the three obvious active site mutations for providing
an oxyanion hole pocket to \textit{Hb}HNL and one additional stabilization
mutation yielded HNL3V, which shows poor esterase activity. The SPM
analysis applied to the reference SABP2 esterase (top) and HNL3V
(bottom) is depicted, with spheres indicating residues shared between
both proteins and cubes marking differing residues. Four/five SPM
positions were selected for mutagenesis, yielding HNL7 and HNL8 that
both exhibit a 300-fold enhancement in esterase catalytic activity as
compared to \textit{Hb}HNL. The incorporation of evolutionary information into
SPM provided HNL7T, which displays a 1450-fold increase in catalytic
efficiency, thus surpassing that of the SABP2 esterase taken as
reference.}
{\vspace*{4pt}}
\end{figure*}

In another recent study also focused on a tryptophan synthase, the SPM
tool was employed to enhance the \textbf{stand-alone activity of Zea
mays TrpA (\textit{Zm}TrpA)} by identifying key
conformationally relevant positions~\cite{79} (Figure~\ref{fig4}b). Unlike TrpB, where a
shift in allosteric regulation was observed along its phylogenetic
tree, LBCA TrpA was already allosterically activated by
TrpB~\cite{77,78}. This made more difficult the identification of
positions for stand-alone activity using traditional methods like
MSA. However, the structurally similar
stand-alone enzyme \textit{Zm}BX1~\cite{80} from the secondary
metabolism of maize served as a model for designing an enhanced TrpA
variant~\cite{79}. 

The initial approach involved engineering the \textit{Zm}TrpA to the
\textit{Zm}TrpA-L6Bx1 variant by switching the L6 loop with the L6
found in ZmBX1~\cite{80}, as L6 closed conformation observed in
\textit{Zm}BX1 was \mbox{reported} to be crucial to TrpA stand-alone
activity~\cite{79}. This resulted in a significant increase in
catalytic turnover ($k_{\mathrm{cat}}$) but at the expense of
increasing K$_{\mathrm{M}}$, indicating the need for additional
improvements to achieve better catalytic efficiency~\cite{80}. To
further enhance the stand-alone activity of \textit{Zm}TrpA, the SPM
method was applied to identify key conformationally relevant positions
by comparing the intramolecular pathways of \textit{Zm}TrpA-L6BX1 and
\textit{Zm}BX1 at the catalytically activated state. The motivation for
specifically targeting the IGP-bound catalytically activated state was
the experimental observation that the conformational transition to
achieve the catalytically active state is rate-determining~\cite{81}.
By comparing the generated SPM for the starting \textit{Zm}TrpA-L6BX1
scaffold and \textit{Zm}BX1, a subset of non-conserved
SPM-identified positions was selected for mutation, leading to the
\textit{Zm}TrpA-SPM4-L6BX1 variant, which exhibits a 178-fold
improvement in catalytic efficiency towards IGP cleavage~\cite{79}.
With this study, the power of SPM in identifying both active site and
distal mutations to boost stand-alone activity is highlighted and paves
the way for enzyme design in those enzymatic systems where
conformational change is rate-limiting. 

The SPM tool was also successfully applied to predict mutations that
converted \textbf{hydroxynitrile lyase (HbHNL)} from the rubber tree into
an \mbox{efficient} esterase by comparison with tobacco \textbf{esterase
(SABP2)}~\cite{82} (Figure~\ref{fig4}c). Both enzymes are present in the
$\upalpha,\upbeta$-hydrolase fold and despite the fact that they share the same
catalytic Ser--His--Asp triad, they exhibit very different forms of reactivity.
Initial efforts to engineer \textit{Hb}HNL into an esterase produced
the HNL3V variant, which focused on the three obvious active site
mutations (T11G, E79H, K236M) to introduce the oxyanion hole residues
and to replace the polar site by a hydrophobic site as observed in
SABP2. However, these three obvious mutations only showed limited
esterase activity. To further enhance the esterase activity, SPM
analysis of HNL3V, a stabilized version of HNL3 for \mbox{experimental}
purposes, and SABP2 revealed five additional positions outside the
active site, which were connected to the catalytic residues and could
potentially affect the oxyanion hole and the catalytic Asp preorganization.
By replacing these positions with those from SABP2, the HNL6V, HNL7V,
and HNL8V variants were created, enhancing esterase activity by
300-fold. The SPM-guided mutations targeted correlated movements
essential for catalysis by removing unwanted movements in HNL3V and
adding the missing ones from SABP2. To this end, the residues in HNL3V
were replaced with the corresponding ones from SABP2. Notably, most of
the engineered mutations were outside the active site, demonstrating
the ability of the SPM to identify distal residues contributing to
catalytic function. 

The combination of SPM analysis with an MSA of esterases resulted in
the identification of the most effective variant, HNL7TV. Among the
mutations tested, the N104A substitution produced the largest increase
in turnover rate ($k_{\mathrm{cat}}$). However, sequence alignment
revealed that most esterases naturally have a threonine at this
position rather than an alanine.
Introducing the N104T substitution led
to the best overall variant with approximately twice the catalytic
efficiency ($k_{\mathrm{cat}}/\mathrm{K}_{\mathrm{M}}$)
of SABP2, thus increasing the esterase catalytic
efficiency of the starting \textit{Hb}HNL scaffold by 1450-fold.
Despite this impressive improvement, the turnover rate
($k_{\mathrm{cat}}$) of HNL7TV remained 13-fold lower than SABP2,
indicating a need for further refinement. Nevertheless, the dramatic
increase in catalytic efficiency demonstrated the ability of the SPM to
predict which residues outside the active site contribute to catalytic
activity~\cite{82}.  

\section{Conclusions}\label{sec3}

Recognizing the high similarity between allosteric processes and the
effect exerted by distal active site mutations on the enzyme catalytic
activity, we hypothesized that the computational identification of
allosteric networks could bring interesting insights into enzymatic
function and design~\cite{44}. To that end, we developed the 
SPM method~\cite{44,52}, which identifies a set of
conformationally relevant positions based on the application of graph
theory to the \mbox{distance} and correlation matrices computed from MD
data. Over the years, we (and others) have applied the SPM to
identify conformationally relevant positions and compared those with
residues mutated in DE
experiments~\cite{2,44,48,58,63}. Interestingly, in many different
enzyme classes and families, the application of SPM has revealed that
some of the DE mutations are introduced in conformationally relevant sites,
thus suggesting its potential application for enzyme design. 

The SPM has also been used for elucidating the intricate mechanisms of
allosteric regulation in different enzymatic and protein
systems~\cite{64,69,74}. The SPM allosteric networks identify critical
pathways of residues, some of which are experimentally known to
have a large effect on  enzyme/protein regulation. We have also
tested the applicability of SPM, especially if combined with 
ASR and/or MSA,
to rationally design new improved enzymes with enhancements in
catalytic efficiency that range from 7- to 1450-fold~\cite{76,79,82}.
Altogether these examples show the potential of the SPM method for
understanding the effect of distal mutations into the enzyme
conformational dynamics and how they affect enzymatic function. More
importantly, integrating SPM with other computational techniques such
as MSA and ASR corresponds to a successful approach for rational enzyme
design. 

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

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