Kinetic network models to elucidate aggregation dynamics of aggregation‐induced emission systems

Aggregation‐induced emission (AIE) is a phenomenon where a molecule that is weakly or non‐luminescent in a diluted solution becomes highly emissive when aggregated. AIE luminogens (AIEgens) hold promise in diverse applications like bioimaging, chemical sensing, and optoelectronics. Investigation in AIE luminescence is also critical for understanding aggregation kinetics as the aggregation process is an essential component of AIE emission. Experimental investigation of AIEgen aggregation is challenging due to the fast timescale of the aggregation and the amorphous aggregate structures. Computer simulations such as molecular dynamics (MD) simulation provide a valuable approach to complement experiments with atomic‐level knowledge to study the fast dynamics of aggregation processes. However, individual simulations still struggle to systematically elucidate heterogeneous kinetics of the formation of amorphous AIEgen aggregates. Kinetic network models (KNMs), constructed from an ensemble of MD simulations, hold great potential in addressing this challenge. In these models, dynamic processes are modeled as a series of Markovian transitions occurring among metastable conformational states at discrete time intervals. In this perspective article, we first review previous studies to characterize the AIEgen aggregation kinetics and their limitations. We then introduce KNMs as a promising approach to elucidate the complex kinetics of aggregations to address these limitations. More importantly, we discuss our perspective on linking the output of KNMs to experimental observations of time‐resolved AIE luminescence. We expect that this approach can validate the computational predictions and provide great insights into the aggregation kinetics for AIEgen aggregates. These insights will facilitate the rational design of improved AIEgens in their applications in biology and materials sciences.

kinetic mechanisms of aggregation and the photophysical mechanisms of emission that occur following the aggregation.The latter has been extensively studied, leading to the identification of several major photophysical mechanisms that explain the AIE emission. [5]The first mechanism involves the emergence of emission resulting from dipoleallowed radiation caused by changes that occur at the electronic states upon aggregation.A notable example is terephthalic acid, which exhibits strong luminescence due to the formation of additional electrostatic interactions in the aggregate phase, rendering the previously 'dark' states luminescent. [6]The second mechanism involves [6] the blockage of non-radiative decay through vibration relaxation (BNR-VR) in harmonic regions.Hexaphenylsilole (HPS) serves as an example of such mechanism as its non-radiative decay pathway is hindered by the restriction of intramolecular vibration motion after aggregation. [7]The third mechanism involves the removal of non-radiation via an isomerization/a minimum energy crossing point (RNR-ISO/RNR-MECP).An example illustrating this mechanism is 4-diethylamino-2-benzylidene malonic acid dimethyl ester, which exhibits a notably distinct excited state structure in the solution and aggregation phases, resulting in different radiative behaviors between these two phases. [8]nvestigation of AIE aggregation kinetics, however, is limited by the spatial resolution and time sensibility of experimental techniques.AIE systems differ from other self-assembly systems because of their fast kinetics (sub-millisecond) and the amorphous nature of AIEgen aggregates. [9]Direct observation of fluorescence upon aggregation using experimental techniques requires the experimental setup to have a temporal resolution on the scale of microseconds.Furthermore, the fluorescence of AIE is weak compared to other forms of light emission, making it difficult for most spectroscopy techniques to accurately quantify the emission intensity.Nevertheless, certain experimental techniques such as microfluidics with fluorescence spectroscopy can be adopted to track the time evolution of the aggregation process.In one study, a microfluidics setup allowed researchers to observe the aggregation process at microsecond resolution, helping them elucidate the aggregation kinetics with some success.However, these techniques cannot provide molecular-level details of the aggregation process. [9]omputer simulations such as MD simulations provide a valuable approach to complement experiments in studying the kinetics of AIE aggregation processes. [10]However, the heterogenous kinetics of AIE aggregation pose a major challenge in understanding the underlying aggregation mechanisms through visual inspection of individual simulation trajectories. [11]Kinetic network models (KNMs) built from an ensemble of MD simulations hold promise in addressing this challenge by providing statistical representations of AIEgen aggregate structures and elucidating the major aggregation pathways. [12]In this approach, the conformational space is partitioned into a set of metastable states, and time is also coarse-grained into discrete units.With a proper selection of the time interval, the continuous dynamics can be simplified as Markovian transitions among different conformational states.[26] In this perspective article, we will commence by reviewing previous studies aimed at modeling the aggregation kinetics of AIE using an analytical nucleation-growth theory. [9]ubsequently, we will demonstrate the application of KNMs constructed from MD simulations to study the kinetics of aggregation.Additionally, we will review recent methodological advancements in KNMs, particularly the utilization of machine learning tools, to enhance the construction of KNMs for studying aggregation processes. [27]Most importantly, we will offer our perspective on enabling quantitative comparisons between KNM predictions and time-resolved AIE emissions measured by experiments.This comparative analysis presents an opportunity to validate theoretical models and facilitate the fundamental understanding of the AIE phenomenon.

ELUCIDATING AIE AGGREGATION KINETICS BY INTEGRATING THE MICROFLUIDICS EXPERIMENT WITH THE NUCLEATION-GROWTH THEORY
The unique aggregation-induced light emission of AIEgens allows the investigation of the aggregation process by monitoring the time-resolved fluorescence intensity. [3]In the work of Jiang et al., a microfluid experimental setup was employed to study the HPS aggregation, in which the rapid mixing (at microseconds) of the HPS solution in dimethylsulfoxide (DMSO) with water triggers the aggregation process (Figure 1B). [9]Driven by the strong hydrophobic interactions, the dispersed HPS molecules immediately start to aggregate along the microfluidics tube upon mixing.By measuring the fluorescence intensity at different locations along the microfluidics tube, Jiang et al. were able to determine the progress of aggregation at microsecond resolution (Figure 1C).
Jiang et al. have further applied the classical nucleationgrowth theory (CNT) [29] to fit the observed time-resolved fluorescence data to obtain the free energies and kinetics of the HPS aggregation.[32] The solvation free energy change between different HPS aggregate sizes reflects the free energy associated with attaching HPS molecules to existing aggregates.In addition, Jiang et al. [29] also calculated the time needed for aggregates to grow by one HPS molecule with this fitted model (Figure 2C).They determined that the minimal time to attach one HPS molecule to existing aggregates in water is around 25-50 ns at a monomer concentration of millimolar.The timescale is in accordance with previous observation of early-stage protein folding, which is driven by hydrophobic interactions. [32]This calculation benefits from the application of microfluidic experimental setup that enabled observation on a microsecond resolution, which is a critical time range for understanding the initial hydrophobic aggregation process.In addition to hydrophobic interactions, the microfluidic device would also be applicable to monitor the self-assembly/aggregation processes driven by other intermolecular interactions (e.g., polar interactions).However, two conditions need to be satisfied: (a) the microfluidic device relies on the rapid mixing of two solvents, and is thus applicable to monitor fast kinetic processes occurring at microseconds or sub-millisecond; and (b) there should exist proper spectroscopic observables to trace the amount of aggregates, such as the fluorescence intensity adopted by Jiang et al. [29] for the HPS aggregation.
Notably, Jiang et al. [29] also reported experimental validation of the crossover in the hydrophobic effect, as predicted by the Lum-Chandler-Weeks (LCW) theory. [33]In the LCW theory, the hydration of small hydrophobic solutes is driven by entropy, as these solutes can be accommodated by simply reordering the water's hydrogen bond network.However, the hydration of large hydrophobic molecules requires breaking water-water hydrogen bonds, and thus leading to an enthalpy-driven process.This shift in water hydrogen bond behavior leads to a crossover of unit area solvation free energy with respect to hydrophobic solute size (see the change in the slope of solvation free energy as the solute size reaches 1-nm radius, as shown in Figure 2B).To our knowledge, this result [29] provides the first experimental verification of the theoretical predictions from the LCW theory. [34]owever, modeling the HPS aggregation using the analytical CNT also relies on several assumptions.First, this CNT model relies on the analytical classical nucleation theory, and it cannot provide molecular-level insights.In particular, the aggregates are all modeled as spheres in CNT, which are inadequate to describe their amorphous structures. [35]econd, the fitting to the CNT model assumes the linear relationship between fluorescence intensity and aggregates' volume, which requires the quantum yield to maintain a constant for different HPS aggregates.The latter assumption has been shown to be valid through QM/MM calculations when aggregates reach the size of 20 and more HPS molecules (Figure 1D), [28] as well as through the experiments using the fluorescence confocal atomic force microscopy. [29]The former assumption can be addressed through the application of atomistic MD simulations, which model the aggregation dynamics at a molecular level.However, the hydrophobic aggregation of HPS molecules is a highly inhomogeneous process.As a result, direct inspections of individual MD trajectories are often insufficient for uncovering major kinetic pathways and comprehending the molecular mechanisms underlying the aggregation.KNMs constructed from an ensemble of MD simulations have great potential to elucidate the kinetic mechanisms of the aggregations of HPS and other AIEgen molecules, driven by hydrophobic interactions.

KINETIC NETWORK MODELS TO ELUCIDATE THE AIEGEN AGGREGATION KINETICS
In this section, we discuss the applications of KNMs in the study of AIEgen aggregation kinetics.Dissecting the kinetic mechanisms of AIEgen aggregations is challenging due to the complexity and heterogeneity of processes driven by nonspecific hydrophobic interactions.MD simulations offer a valuable approach to complement experiments for studying the kinetics of AIEgen aggregation.However, it remains challenging to elucidate the complex kinetic mechanisms underlying AIEgen aggregation using individual MD simulation trajectories.KNMs hold promise in addressing this challenge and revealing the kinetic mechanisms of AIEgen aggregation.KNMs are an approach used for computing the thermodynamic and kinetic properties of a kinetic process from an ensemble of MD simulations. [12,13,36,37]In KNMs, the configuration space is partitioned into a set of metastable states, and time is also coarse-grained into discrete units.With a proper selection of the time interval, the continuous dynamics can be simplified as Markovian transitions among different conformational states. [18,38]The Transition Path Theory (TPT) [39] can then be applied to identify transition pathways between a pair of states, and further characterize their properties such as the dominant pathways and the flux of individual ones.
A general pipeline to build a KNM is summarized as follows (Figure 3) [13,37,38,40] : (i) conduct MD simulations initiated from different conformations and obtain an ensemble of MD simulation trajectories of the aggregation process [41] (Figure 3A); (ii) select geometric features [12] to describe the aggregate conformations, and further perform dimensionality reduction [42,43] using tICA [44] and VAMPnet/SRVnets [27,45,46] to find the slow collective variables (CVs) that can describe the kinetics of the aggregation processes; (iii) cluster the MD conformations into microstates (e.g., k-centers, [47,48] k-means, [49] etc.) according to the pairwise distances defined as Euclidian distances in the CV space (Figure 3B); (iv) lump microstates (usually hundreds to thousands) into a handful of metastable macrostates based on their kinetic connectivity using kinetic lumping algorithms, such as PCCA and PCCA+ [50][51][52] ; (v) build the KNM (Figure 3C), which contains a transition probability matrix (TPM).26] This mechanistic understanding can facilitate the rational design of the monomer chemical structures to produce specific self-assembled structures of interest in the long-term.For example, Zheng et al. [26] successfully applied KNMs to reveal distinct kinetic mechanisms for the self-assembly of two amphophilic molecules that only slightly differ in their chemical structures (i.e., PYR and PYN).In particular, they identified two coexisting kinetic pathway channels: the incremental growth channel, in which the aggregates grow incrementally by absorbing one monomer at a time, and the hopping growth channel, where aggregates with similar sizes merge to form larger aggregates.Interestingly, they found that PYR prefers the incremental growth channel, resulting in the formation of a nanotube, whereas PYN favors a hopping growth channel, leading to the creation of a vesicle. [26]hese results demonstrated the feasibility of rational design of the self-assembled/aggregate structures via the kinetic control (i.e., the selection of specific kinetic pathway) through the fine-tuning of monomer chemical structures.
While KNMs have been proven to be a powerful tool for studying conformational dynamics, [24,26,[57][58][59][60] there are still challenges that hinder their application to aggregation processes. [12]First, conformations in aggregation systems usually consist of identical monomers, which require the feature representation to be invariant to permutations.As a result, typical methods such as tICA [44] that take pairwise distances as representative input features of the analysis may fail.One possible solution is to directly adopt representative physical coordinates that are invariant to permutations as CVs, such as the asphericity parameter A p that describes the morphology of aggregates. [26]However, the identification of those representative physical coordinates requires additional physical insights and is also system specific.Second, the kinetic networks for aggregation systems usually consist of numerous parallel pathways, leading to an overwhelming number of pathways with comparable flux in the direct application of TPT. [61]To address this challenge, Meng et al. developed a path lumping algorithm to deal with this issue, lumping kinetically well-connected pathways into several dominant path channels based on the kinetic similarities (defined as the accumulated flux intercrossing the initial and final states), which helped in the comprehension of aggregation mechanisms. [56]Recently, Qiu et al. [62] have developed a latent space clustering (LPC) machine-learning approach that utilizes the variational autoencoder (VAE) to learn the spatial distributions of TPT kinetic pathways and to perform path clustering in the latent space.The LPC method is demonstrated to outperform the path lumping algorithm, [56] and can efficiently group parallel kinetic pathways into distinct metastable path channels.Third, the aggregation is often an energetically downhill process, rendering dissociations as rare events that are challenging to sample adequately.This often leads to an asymmetric transition count matrix that violates the detailed balance, necessitating equal association and dissociation transition counts.To address this issue, Zeng et al. replaced the equilibrium flux with mass flow as the input for TPT and successfully elucidated the dominant pathways of self-assembly for star-like block copolymers. [24]n recent years, KNMs have witnessed significant advancements, attributed to the rapid development and innovation of machine learning approaches.Mardt et al. developed VAMPnets, [27] which employs the variational approach for Markov processes (VAMP) [46] with deep neural networks.Building upon two neural network lobes for a lagged time interval (), VAMPnets maximizes the linearity of conformation projections for time t and t +  to find the best projections and perform the metastable-state assignment.The nonlinear structure of deep neural networks enables VAMPnets to map the original MD conformations to fuzzy state assignments with high accuracy (i.e., a given MD conformation has the probabilities to be assigned to multiple states).VAMPnets also offers an end-to-end solution for the KNM construction, thus greatly simplifies the application of KNMs to study complex conformational dynamics.[65] Graph neural networks (GNN) represent inputs with nodes and edges that are permutation-invariant by construction.It has also been shown that [66] the learned embeddings of graphs conserve useful information about system's dynamics, making it an efficient way for feature representation for the aggregation process.Combining GNN with VAMPnets, GraphVAMPnets has been developed and successfully applied to study various chemical and biological processes, including ion transition, protein folding, and molecular self-assembly. [63,64,67]nheriting the permutation invariant nature of GNN, Graph-VAMPnets holds great potential to be applied to identify CVs for the AIEgen aggregation process (Figure 4).Furthermore, a GNN trained on one system also has the potential to be transferable for searching CVs in other related systems.For example, a GNN trained for a colloidal system has been successfully transferred to find CVs to study the copper melt crystallization process. [66]n addition to revealing the mechanisms underlying the aggregation process, deep-learning methods can also assist in searching for optimal synthesis protocols based on kinetic information.Whitelam et al. [68] combined reinforcement learning (RL) with Monte Carlo simulations to find the optimal change of system-controlled parameters to obtain highly ordered self-assembly aggregates.In their model, molecules receive an energetic reward if their center-tocenter distance lies within the range of bonding, and models with the highest polygon counts are prioritized as effective steps toward ordered structures.This approach establishes a control policy for generating highly ordered structures, which has been demonstrated to be more efficient than previously known control protocols.In addition, their model is capable of performing an optimal control policy search for desired structures, thereby offering valuable physical insights into the aggregation process.Tang et al. [69] employed reinforcement learning to develop a feedback-control-based method that effectively guides the colloidal assembly dynamics toward perfect crystal structures.They built multiple KNMs under different conditions, such as varying external magnetic field.By leveraging transition probabilities from KNMs, this method conducts optimal control policy searches and effectively steers the system toward the intended perfect crystal structure.Similarly, Ma and Ferguson [70] utilized evolutionary algorithms to iteratively shape the free energy surface, with the aim of identifying the optimal free energy surface that enables efficient transitions to desired aggregate structures.This optimal free energy surface is subsequently translated into a set of design parameters (e.g., temperature, interaction strength, and polar angles) to achieve aggregation with high precision.In the future, we anticipate that these reinforcement learning approaches could also offer advantages for investigating AIEgen aggregation.

PERSPECTIVE ON KNM MODELING OF TIME EVOLUTION OF AIE FLUORESCENCE
Despite the advantages of applying KNMs to aggregation systems, there are rarely quantitative comparisons between the results from KNMs and experiments, which are important to validate KNMs.However, the time evolution of state populations predicted by KNMs can't be directly observed by experiments.To make the connection, we can compute The schematic structure of GraphVAMPnets for self-assembly system.(A) Conformations of assembly system and their time-lagged counterparts are extracted from trajectories.(B) These conformations are then embedded by graphs and fed into two parallel graph convolutional neural networks.(C) The latent representation of each conformation is obtained through graph pooling followed by the VAMPnet optimization.This figure is reproduced from Ref. [63]   the time evolution of fluorescence intensity based on the fluorescence intensities of different states. [28]This approach provides a quantitative evaluation of the accordance between results from experiments and KNMs.
A similar concept of quantitative comparison was previously employed in the study by Zhuang et al., [71] where they compared the predictions from KNMs with the infrared (IR) and 2DIR spectroscopy.Specifically, they studied the trpzip2 peptide unfolding process triggered by the temperature jump (T-jump) induced by laser.To model this unfolding process, Zhuang et al. [71] first constructed two KNMs, one at the temperature before the T-jump and another after the T-jump.After the T-jump, they calculated the relaxation of state populations by propagating the TPM at the higher temperature, while initializing the state populations with the stationary populations at the lower temperature before the Tjump.For each metastable state, they computed the FTIR and 2DIR spectra signals using the SPECTRON [72] and NEP [73] packages.This enabled them to calculate the time evolution of the FTIR/2DIR spectra following the T-jump, achieved by averaging spectra signals from different states weighted by their populations at each time point.They demonstrated good agreement between the calculated spectra signals from KNMs and experimental observations. [71]hen considering AIEgen aggregation, one can compare the time-resolved fluorescence intensity between KNM's predictions and the experimental observations from a previous study. [29]After successfully constructing a KNM through extensive MD simulations of the HPS aggregation, the propagation of its TPM can predict the time evolution of state populations with an initial state distribution, in which all the MD conformations are in the dispersed state containing only monomers (Figure 5B).For every metastable state within this KNM, the AIE fluorescence intensity can be estimated using QM/MM calculations on representative conformations from each state, following a well-established protocol. [28]By averaging the fluorescence intensity across different states, weighted by their respective populations at each time point, one can derive the time evolution of the overall fluorescence intensity.The calculated time evolution of fluorescence intensity can subsequently be directly compared to the experimentally observed fluorescence intensity in the microfluidics experiment (Figure 5C).Performing quantitative comparisons between KNM's predictions and experiments will enhance the understanding of the kinetics of AIEgen aggregations at a molecular resolution, a detail that is often absent in spectroscopy experiments.Consequently, these quantitative comparisons will also contribute to refining the theoretical model.To monitor the time evolution of AIE aggregate formation, one needs to apply a specific triggering condition, such as rapidly mixing an organic solvent with water to induce hydrophobic aggregation in the microfluidic device. [29]In the absence of triggering events, aggregates will form simultaneously, and one can only measure the emission intensity of the entire aggregate ensemble at equilibrium.Under this condition, KNMs can still be applied to compute the equilibrium emission spectra.This entails calculating the average emission intensity of different metastable aggregate states, weighted by their equilibrium conditions obtained from KNMs.

CONCLUSIONS
In this perspective article, we have demonstrated the application of KNMs constructed from MD simulations to study the kinetics of AIEgen aggregation.In recent years, KNMs have undergone significant advancements, driven by the rapid development and innovation of machine learning approaches.
These methodological enhancements hold the potential to greatly enhance the application of KNMs to AIEgen aggregation processes.Regarding the comparison between theory and experiment, we have introduced an approach that facilitates quantitative comparisons between KNM's predictions and time-resolved AIE emissions measured in experiments.We anticipate that this comparative analysis can validate theoretical models and contribute to a deeper understanding of the AIE phenomenon.On the experimental front, we foresee the potential application of techniques such as photochemical triggering (photodissociation or photo-induced electron transfer reactions) and stopped-flow techniques to induce AIEgen aggregation, in addition to microfluidics.Furthermore, spectroscopic signals beyond fluorescence intensity could be harnessed to monitor aggregation kinetics.Notably, a recent study has employed UV absorption induced by metal-metal-to-ligand charge transfer in a platinum (II) complex to monitor its self-assembly dynamics. [74]In summary, we expect that the integration of theoretical KNMs with experimental approaches could yield profound insights into the aggregation kinetics of AIEgen aggregates.These insights will facilitate the rational design of improved AIEgen molecules for applications in biology and materials sciences.

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I G U R E 1 (A) Free energy change and cooperativity associated with HPS aggregation in aqueous solution.(B) Experimental setup of microfluidic mixing: HPS in DMSO solution is mixed with water at the intersection of the microfluidic tube.Solvent exchange completes within microseconds, followed by HPS aggregation, monitored by fluorescence intensity from AIE. (C) Fluorescence intensity as a function of time with different initial HPS concentrations.The dots represent experimental data, and the lines depict predictions from the classical nucleation-growth theory (CNT) theory.(D) Left: Two snapshots of amorphous aggregates with the HPS molecule of interest positioned at the center ("embedded", top panel) and on the surface of the aggregate ("exposed", bottom panel).Right: Calculated fluorescence quantum efficiency as a function of aggregate size.Panels A-C are reproduced from Ref. [9], while panel D is reproduced from Ref. [28] F I G U R E 2 (A) Illustration of the classical nucleation-growth (CNT) theory.(B) Crossover in solvation free energy per surface area versus the size of aggregate.(C) Time needed for adding one HPS molecule to existing aggregate of different sizes.Results of the system with 16%, 21%, 26%, and 32% of DMSO are shown in black, purple, cyan, and yellow, respectively.The figure is reproduced from Ref. [9]

F I G U R E 3
Construction of the Kinetic Network Model (KNM) for the aggregation.(A) Illustration of the aggregation process.(B) Projection of MD conformations onto the CV space.(C) Visualization of metastable macrostate in the CV space and the associated transition probabilities between them (indicated by white arrows).(D) The resulting kinetic network representation.

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I G U R E 5 (A) KNMs for the AIEgen aggregation.(B) Prediction of time evolution of state populations starting from the monomer state.Darker colors indicate larger population.(C) Calculated time evolution of fluorescence intensity.
Xuhui Huang acknowledges the support from the Office of the Vice-Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation and the Hirschfelder Professorship Fund.C O N F L I C T O F I N T E R E S T S TAT E M E N TThe authors declare they have no conflicts of interest.O R C I DBojun Liu https://orcid.org/0009-0007-3960-1910Xuhui Huang https://orcid.org/0000-0002-7119-9358RE F E R E N C E S