Pairwise Maximum Entropy Model Research Articles

Energy of functional brain states correlates with cognition in adolescent-onset schizophrenia and healthy persons.

Adolescent-onset schizophrenia (AOS) is rare, under-studied, and associated with more severe cognitive impairments and poorer outcomes than adult-onset schizophrenia. Neuroimaging has shown altered regional activations (first-order effects) and functional connectivity (second-order effects) in AOS compared to controls. The pairwise maximum entropy model (MEM) integrates first- and second-order factors into a single quantity called energy, which is inversely related to probability of occurrence of brain activity patterns. We take a combinatorial approach to study multiple brain-wide MEMs of task-associated components; hundreds of independent MEMs for various sub-systems are fit to 7 Tesla functional MRI scans. Acquisitions were collected from 23 AOS individuals and 53 healthy controls while performing the Penn Conditional Exclusion Test (PCET) for executive function, which is known to be impaired in AOS. Accuracy of PCET performance was significantly reduced among AOS compared to controls. A majority of the models showed significant negative correlation between PCET scores and the total energy attained over the fMRI. Across all instantiations, the AOS group was associated with significantly more frequent occurrence of states of higher energy, assessed with a mixed effects model. An example MEM instance was investigated further using energy landscapes, which visualize high and low energy states on a low-dimensional plane, and trajectory analysis, which quantify the evolution of brain states throughout this landscape. Both supported patient-control differences in the energy profiles. Severity of psychopathology was correlated positively with energy. The MEM integrated representation of energy in task-associated systems can help characterize pathophysiology of AOS, cognitive impairments, and psychopathology.

The alterations of repetitive transcranial magnetic stimulation on the energy landscape of resting-state networks differ across the human cortex.

Repetitive transcranial magnetic stimulation (rTMS) is a promising intervention tool for the noninvasive modulation of brain activity and behavior in neuroscience research and clinical settings. However, the resting-state dynamic evolution of large-scale functional brain networks following rTMS has rarely been investigated. Here, using resting-state fMRI images collected from 23 healthy individuals before (baseline) and after 1 Hz rTMS of the left frontal (FRO) and occipital (OCC) lobes, we examined the different effects of rTMS on brain dynamics across the human cortex. By fitting a pairwise maximum entropy model (pMEM), we constructed an energy landscape for the baseline and poststimulus conditions by fitting a pMEM. We defined dominant brain states (local minima) in the energy landscape with synergistic activation and deactivation patterns of large-scale functional networks. We calculated state dynamics including appearance probability, transitions and duration. The results showed that 1 Hz rTMS induced increased and decreased state probability, transitions and duration when delivered to the FRO and OCC targets, respectively. Most importantly, the shortest path and minimum cost between dominant brain states were altered after stimulation. The absolute sum of the costs from the source states to the destinations was lower after OCC stimulation than after FRO stimulation. In conclusion, our study characterized the dynamic trajectory of state transitions in the energy landscape and suggested that local rTMS can induce significant dynamic perturbation involving stimulated and distant functional networks, which aligns with the modern view of the dynamic and complex brain. Our results suggest low-dimensional mapping of rTMS-induced brain adaption, which will contribute to a broader and more effective application of rTMS in clinical settings.

Spontaneous Brain Activity Emerges from Pairwise Interactions in the Larval Zebrafish Brain

Brain activity is characterized by brainwide spatiotemporal patterns that emerge from synapse-mediated interactions between individual neurons. Calcium imaging provides access to recordings of whole-brain activity at single-neuron resolution and, therefore, allows the study of how large-scale brain dynamics emerge from local activity. In this study, we use a statistical mechanics approach—the pairwise maximum entropy model—to infer microscopic network features from collective patterns of activity in the larval zebrafish brain and relate these features to the emergence of observed whole-brain dynamics. Our findings indicate that the pairwise interactions between neural populations and their intrinsic activity states are sufficient to explain observed whole-brain dynamics. In fact, the pairwise relationships between neuronal populations estimated with the maximum entropy model strongly correspond to observed structural connectivity patterns. Model simulations also demonstrated how tuning pairwise neuronal interactions drives transitions between observed physiological regimes and pathologically hyperexcitable whole-brain regimes. Finally, we use virtual resection to identify the brain structures that are important for maintaining the brain in a physiological dynamic regime. Together, our results indicate that whole-brain activity emerges from a complex dynamical system that transitions between basins of attraction whose strength and topology depend on the connectivity between brain areas. Published by the American Physical Society 2024

An energy landscape approach reveals the potential key bacteria contributing to the development of inflammatory bowel disease.

The dysbiosis of microbiota has been reported to be associated with numerous human pathophysiological processes, including inflammatory bowel disease (IBD). With advancements in high-throughput sequencing, various methods have been developed to study the alteration of microbiota in the development and progression of diseases. However, a suitable approach to assess the global stability of the microbiota in disease states through time-series microbiome data is yet to be established. In this study, we have introduced a novel Energy Landscape construction method, which incorporates the Latent Dirichlet Allocation (LDA) model and the pairwise Maximum Entropy (MaxEnt) model for their complementary advantages, and demonstrate its utility by applying it to an IBD time-series dataset. Through this approach, we obtained the microbial assemblages' energy profile of the whole microbiota under the IBD condition and uncovered the hidden stable stages of microbiota structure during the disease development with time-series microbiome data. The Bacteroides-dominated assemblages presenting in multiple stable states suggest the potential contribution of Bacteroides and interactions with other microbial genera, like Alistipes, and Faecalibacterium, to the development of IBD. Our proposed method provides a novel and insightful tool for understanding the alteration and stability of the microbiota under disease states and offers a more holistic view of the complex dynamics at play in microbiota-mediated diseases.

Diagnostically distinct resting state fMRI energy distributions: A subject-specific maximum entropy modeling study.

Objective Existing neuroimaging studies of psychotic and mood disorders have reported brain activation differences (first-order properties) and altered pairwise correlation based functional connectivity (second-order properties). However, both approaches have certain limitations that can be overcome by integrating them in a pairwise maximum entropy model (MEM) that better represents a comprehensive picture of fMRI signal patterns and provides a system-wide summary measure called energy. This study examines the applicability of individual-level MEM for psychiatry and identifies image-derived model coefficients related to model parameters. Method MEMs are fit to resting state fMRI data from each individual with schizophrenia/schizoaffective disorder, bipolar disorder, and major depression (n=132) and demographically matched healthy controls (n=132) from the UK Biobank to different subsets of the default mode network (DMN) regions. Results The model satisfactorily explained observed brain energy state occurrence probabilities across all participants, and model parameters were significantly correlated with image-derived coefficients for all groups. Within clinical groups, averaged energy level distributions were higher in schizophrenia/schizoaffective disorder but lower in bipolar disorder compared to controls for both bilateral and unilateral DMN. Major depression energy distributions were higher compared to controls only in the right hemisphere DMN. Conclusions Diagnostically distinct energy states suggest that probability distributions of temporal changes in synchronously active nodes may underlie each diagnostic entity. Subject-specific MEMs allow for factoring in the individual variations compared to traditional group-level inferences, offering an improved measure of biologically meaningful correlates of brain activity that may have potential clinical utility.

The quality and complexity of pairwise maximum entropy models for large cortical populations.

We investigate the ability of the pairwise maximum entropy (PME) model to describe the spiking activity of large populations of neurons recorded from the visual, auditory, motor, and somatosensory cortices. To quantify this performance, we use (1) Kullback-Leibler (KL) divergences, (2) the extent to which the pairwise model predicts third-order correlations, and (3) its ability to predict the probability that multiple neurons are simultaneously active. We compare these with the performance of a model with independent neurons and study the relationship between the different performance measures, while varying the population size, mean firing rate of the chosen population, and the bin size used for binarizing the data. We confirm the previously reported excellent performance of the PME model for small population sizes N < 20. But we also find that larger mean firing rates and bin sizes generally decreases performance. The performance for larger populations were generally not as good. For large populations, pairwise models may be good in terms of predicting third-order correlations and the probability of multiple neurons being active, but still significantly worse than small populations in terms of their improvement over the independent model in KL-divergence. We show that these results are independent of the cortical area and of whether approximate methods or Boltzmann learning are used for inferring the pairwise couplings. We compared the scaling of the inferred couplings with N and find it to be well explained by the Sherrington-Kirkpatrick (SK) model, whose strong coupling regime shows a complex phase with many metastable states. We find that, up to the maximum population size studied here, the fitted PME model remains outside its complex phase. However, the standard deviation of the couplings compared to their mean increases, and the model gets closer to the boundary of the complex phase as the population size grows.

Aberrant brain dynamics in individuals with clinical high risk of psychosis.

Resting-state network (RSN) functional connectivity analyses have profoundly influenced our understanding of the pathophysiology of psychoses and their clinical high risk (CHR) states. However, conventional RSN analyses address the static nature of large-scale brain networks. In contrast, novel methodological approaches aim to assess the momentum state and temporal dynamics of brain network interactions. Fifty CHR individuals and 33 healthy controls (HC) completed a resting-state functional MRI scan. We performed an Energy Landscape analysis, a data-driven method using the pairwise maximum entropy model, to describe large-scale brain network dynamics such as duration and frequency of, and transition between, different brain states. We compared those measures between CHR and HC, and examined the association between neuropsychological measures and neural dynamics in CHR. Our main finding is a significantly increased duration, frequency, and higher transition rates to an infrequent brain state with coactivation of the salience, limbic, default mode and somatomotor RSNs in CHR as compared to HC. Transition of brain dynamics from this brain state was significantly correlated with processing speed in CHR. In CHR, temporal brain dynamics are attracted to an infrequent brain state, reflecting more frequent and longer occurrence of aberrant interactions of default mode, salience, and limbic networks. Concurrently, more frequent and longer occurrence of the brain state is associated with core cognitive dysfunctions, predictors of future onset of full-blown psychosis.

Higher-order interaction learning of line failure cascading in power networks

Line failure cascading in power networks is a complex process that involves direct and indirect interactions between lines’ states. We consider the inverse problem of learning statistical models to find the sparse interaction graph from the pairwise statistics collected from line failures data in the steady states and over time. We show that the weighted l1-regularized pairwise maximum entropy models successfully capture pairwise and indirect higher-order interactions undistinguished by observing the pairwise statistics. The learned models reveal asymmetric, strongly positive, and negative interactions between the network’s different lines’ states. We evaluate the predictive performance of models over independent trajectories of failure unfolding in the network. The static model captures the failures’ interactions by maximizing the log-likelihood of observing each link state conditioned to other links’ states near the steady states. We use the learned interactions to reconstruct the network’s steady states using the Glauber dynamics, predicting the cascade size distribution, inferring the co-susceptible line groups, and comparing the results against the data. The dynamic interaction model is learned by maximizing the log-likelihood of the network’s state in state trajectories and can successfully predict the network state for failure propagation trajectories after an initial failure.

Empirical Bayes estimation of pairwise maximum entropy model for nonlinear brain state dynamics

The pairwise maximum entropy model (pMEM) has recently gained widespread attention to exploring the nonlinear characteristics of brain state dynamics observed in resting-state functional magnetic resonance imaging (rsfMRI). Despite its unique advantageous features, the practical application of pMEM for individuals is limited as it requires a much larger sample than conventional rsfMRI scans. Thus, this study proposes an empirical Bayes estimation of individual pMEM using the variational expectation-maximization algorithm (VEM-MEM). The performance of the VEM-MEM is evaluated for several simulation setups with various sample sizes and network sizes. Unlike conventional maximum likelihood estimation procedures, the VEM-MEM can reliably estimate the individual model parameters, even with small samples, by effectively incorporating the group information as the prior. As a test case, the individual rsfMRI of children with attention deficit hyperactivity disorder (ADHD) is analyzed compared to that of typically developed children using the default mode network, executive control network, and salient network, obtained from the Healthy Brain Network database. We found that the nonlinear dynamic properties uniquely established on the pMEM differ for each group. Furthermore, pMEM parameters are more sensitive to group differences and are better associated with the behavior scores of ADHD compared to the Pearson correlation-based functional connectivity. The simulation and experimental results suggest that the proposed method can reliably estimate the individual pMEM and characterize the dynamic properties of individuals by utilizing empirical information of the group brain state dynamics.

Divergence in Population Coding for Space between Dorsal and Ventral CA1.

Molecular, anatomic, and behavioral studies show that the hippocampus is structurally and functionally heterogeneous, with dorsal hippocampus implicated in mnemonic processes and spatial navigation and ventral hippocampus involved in affective processes. By performing electrophysiological recordings of large neuronal populations in dorsal and ventral CA1 in head-fixed mice navigating a virtual environment, we found that this diversity resulted in different strategies for population coding of space. Populations of neurons in dorsal CA1 showed more complex patterns of activity, which resulted in a higher dimensionality of neural representations that translated to more information being encoded, as compared ensembles in vCA1. Furthermore, a pairwise maximum entropy model was better at predicting the structure of these global patterns of activity in ventral CA1 as compared with dorsal CA1. Taken together, the different coding strategies we uncovered likely emerge from anatomic and physiological differences along the longitudinal axis of hippocampus and that may, in turn, underpin the divergent ethological roles of dorsal and ventral CA1.

Bayesian estimation of maximum entropy model for individualized energy landscape analysis of brain state dynamics.

The pairwise maximum entropy model (MEM) for resting state functional MRI (rsfMRI) has been used to generate energy landscape of brain states and to explore nonlinear brain state dynamics. Researches using MEM, however, has mostly been restricted to fixed‐effect group‐level analyses, using concatenated time series across individuals, due to the need for large samples in the parameter estimation of MEM. To mitigate the small sample problem in analyzing energy landscapes for individuals, we propose a Bayesian estimation of individual MEM using variational Bayes approximation (BMEM). We evaluated the performances of BMEM with respect to sample sizes and prior information using simulation. BMEM showed advantages over conventional maximum likelihood estimation in reliably estimating model parameters for individuals with small sample data, particularly utilizing the empirical priors derived from group data. We then analyzed individual rsfMRI of the Human Connectome Project to show the usefulness of MEM in differentiating individuals and in exploring neural correlates for human behavior. MEM and its energy landscape properties showed high subject specificity comparable to that of functional connectivity. Canonical correlation analysis identified canonical variables for MEM highly associated with cognitive scores. Inter‐individual variations of cognitive scores were also reflected in energy landscape properties such as energies, occupation times, and basin sizes at local minima. We conclude that BMEM provides an efficient method to characterize dynamic properties of individuals using energy landscape analysis of individual brain states.

A Computational Framework for Controlling the Self-Restorative Brain Based on the Free Energy and Degeneracy Principles.

The brain is a non-linear dynamical system with a self-restoration process, which protects itself from external damage but is often a bottleneck for clinical treatment. To treat the brain to induce the desired functionality, formulation of a self-restoration process is necessary for optimal brain control. This study proposes a computational model for the brain's self-restoration process following the free-energy and degeneracy principles. Based on this model, a computational framework for brain control is established. We posited that the pre-treatment brain circuit has long been configured in response to the environmental (the other neural populations') demands on the circuit. Since the demands persist even after treatment, the treated circuit's response to the demand may gradually approximate the pre-treatment functionality. In this framework, an energy landscape of regional activities, estimated from resting-state endogenous activities by a pairwise maximum entropy model, is used to represent the pre-treatment functionality. The approximation of the pre-treatment functionality occurs via reconfiguration of interactions among neural populations within the treated circuit. To establish the current framework's construct validity, we conducted various simulations. The simulations suggested that brain control should include the self-restoration process, without which the treatment was not optimal. We also presented simulations for optimizing repetitive treatments and optimal timing of the treatment. These results suggest a plausibility of the current framework in controlling the non-linear dynamical brain with a self-restoration process.

Divergence in Population Coding for Space between Dorsal and Ventral CA1

Molecular, anatomic, and behavioral studies show that the hippocampus is structurally and functionally heterogeneous, with dorsal hippocampus implicated in mnemonic processes and spatial navigation and ventral hippocampus involved in affective processes. By performing electrophysiological recordings of large neuronal populations in dorsal and ventral CA1 in head-fixed mice navigating a virtual environment, we found that this diversity resulted in different strategies for population coding of space. We found that the populations of neurons in dorsal CA1 had a higher dimensionality and showed more complex patterns of activity, translating to more information being encoded, as compared ensembles in vCA1. Furthermore, a pairwise maximum entropy model was better at predicting the structure of global patterns of activity in ventral CA1 as compared to dorsal CA1. Taken together, we uncovered different coding strategies that likely emerge from anatomical and physiological differences along the longitudinal axis of hippocampus and that may, in turn, underpin the divergent ethological roles of dorsal and ventral CA1.

Pairwise maximum entropy model explains the role of white matter structure in shaping emergent co-activation states

A major challenge in neuroscience is determining a quantitative relationship between the brain’s white matter structural connectivity and emergent activity. We seek to uncover the intrinsic relationship among brain regions fundamental to their functional activity by constructing a pairwise maximum entropy model (MEM) of the inter-ictal activation patterns of five patients with medically refractory epilepsy over an average of ~14 hours of band-passed intracranial EEG (iEEG) recordings per patient. We find that the pairwise MEM accurately predicts iEEG electrodes’ activation patterns’ probability and their pairwise correlations. We demonstrate that the estimated pairwise MEM’s interaction weights predict structural connectivity and its strength over several frequencies significantly beyond what is expected based solely on sampled regions’ distance in most patients. Together, the pairwise MEM offers a framework for explaining iEEG functional connectivity and provides insight into how the brain’s structural connectome gives rise to large-scale activation patterns by promoting co-activation between connected structures.

Detection of functional connectivity in the brain during visuo-guided grip force tracking tasks: A functional near-infrared spectroscopy study.

The functional connectivity (FC) between multiple brain regions during tasks is currently gradually being explored with functional near-infrared spectroscopy (fNIRS). However, the FC present during grip force tracking tasks performed under visual feedback remains unclear. In the present study, we used fNIRS to measure brain activity during resting states and grip force tracking tasks at 25%, 50%, and 75% of maximum voluntary contraction (MVC) in 11 healthy subjects, and the activity was measured from four target brain regions: the left prefrontal cortex (lPFC), right prefrontal cortex (rPFC), left sensorimotor cortex (lSMC), and right sensorimotor cortex (rSMC). We determined the FC between these regions utilizing three different methods: Pearson's correlation method, partial correlation method, and a pairwise maximum entropy model (MEM). The results showed that the FC of lSMC-rSMC and lPFC-rPFC (interhemispheric homologous pairs) were significantly stronger than those of other brain region pairs. Moreover, FC of lPFC-rPFC was strengthened during the 75% MVC task compared to the other task states and the resting states. The FC of lSMC-lPFC and rSMC-rPFC (intrahemispheric region pairs) strengthened with a higher task load. The results provided new insights into the FC between brain regions during visuo-guided grip force tracking tasks.

Energy landscape of resting magnetoencephalography reveals fronto-parietal network impairments in epilepsy.

Juvenile myoclonic epilepsy (JME) is a form of idiopathic generalized epilepsy. It is yet unclear to what extent JME leads to abnormal network activation patterns. Here, we characterized statistical regularities in magnetoencephalograph (MEG) resting-state networks and their differences between JME patients and controls by combining a pairwise maximum entropy model (pMEM) and novel energy landscape analyses for MEG. First, we fitted the pMEM to the MEG oscillatory power in the front-oparietal network (FPN) and other resting-state networks, which provided a good estimation of the occurrence probability of network states. Then, we used energy values derived from the pMEM to depict an energy landscape, with a higher energy state corresponding to a lower occurrence probability. JME patients showed fewer local energy minima than controls and had elevated energy values for the FPN within the theta, beta, and gamma bands. Furthermore, simulations of the fitted pMEM showed that the proportion of time the FPN was occupied within the basins of energy minima was shortened in JME patients. These network alterations were highlighted by significant classification of individual participants employing energy values as multivariate features. Our findings suggested that JME patients had altered multistability in selective functional networks and frequency bands in the fronto-parietal cortices.

Phylogenetic correlations can suffice to infer protein partners from sequences.

Determining which proteins interact together is crucial to a systems-level understanding of the cell. Recently, algorithms based on Direct Coupling Analysis (DCA) pairwise maximum-entropy models have allowed to identify interaction partners among paralogous proteins from sequence data. This success of DCA at predicting protein-protein interactions could be mainly based on its known ability to identify pairs of residues that are in contact in the three-dimensional structure of protein complexes and that coevolve to remain physicochemically complementary. However, interacting proteins possess similar evolutionary histories. What is the role of purely phylogenetic correlations in the performance of DCA-based methods to infer interaction partners? To address this question, we employ controlled synthetic data that only involve phylogeny and no interactions or contacts. We find that DCA accurately identifies the pairs of synthetic sequences that share evolutionary history. While phylogenetic correlations confound the identification of contacting residues by DCA, they are thus useful to predict interacting partners among paralogs. We find that DCA performs as well as phylogenetic methods to this end, and slightly better than them with large and accurate training sets. Employing DCA or phylogenetic methods within an Iterative Pairing Algorithm (IPA) allows to predict pairs of evolutionary partners without a training set. We further demonstrate the ability of these various methods to correctly predict pairings among real paralogous proteins with genome proximity but no known direct physical interaction, illustrating the importance of phylogenetic correlations in natural data. However, for physically interacting and strongly coevolving proteins, DCA and mutual information outperform phylogenetic methods. We finally discuss how to distinguish physically interacting proteins from proteins that only share a common evolutionary history.

Graph-theoretical analysis for energy landscape reveals the organization of state transitions in the resting-state human cerebral cortex.

The resting-state brain is often considered a nonlinear dynamic system transitioning among multiple coexisting stable states. Despite the increasing number of studies on the multistability of the brain system, the processes of state transitions have rarely been systematically explored. Thus, we investigated the state transition processes of the human cerebral cortex system at rest by introducing a graph-theoretical analysis of the state transition network. The energy landscape analysis of brain state occurrences, estimated using the pairwise maximum entropy model for resting-state fMRI data, identified multiple local minima, some of which mediate multi-step transitions toward the global minimum. The state transition among local minima is clustered into two groups according to state transition rates and most inter-group state transitions were mediated by a hub transition state. The distance to the hub transition state determined the path length of the inter-group transition. The cortical system appeared to have redundancy in inter-group transitions when the hub transition state was removed. Such a hub-like organization of transition processes disappeared when the connectivity of the cortical system was altered from the resting-state configuration. In the state transition, the default mode network acts as a transition hub, while coactivation of the prefrontal cortex and default mode network is captured as the global minimum. In summary, the resting-state cerebral cortex has a well-organized architecture of state transitions among stable states, when evaluated by a graph-theoretical analysis of the nonlinear state transition network of the brain.

Ensemble inhibition and excitation in the human cortex: An Ising-model analysis with uncertainties.

The pairwise maximum entropy model, also known as the Ising model, has been widely used to analyze the collective activity of neurons. However, controversy persists in the literature about seemingly inconsistent findings, whose significance is unclear due to lack of reliable error estimates. We therefore develop a method for accurately estimating parameter uncertainty based on random walks in parameter space using adaptive Markov-chain Monte Carlo after the convergence of the main optimization algorithm. We apply our method to the activity patterns of excitatory and inhibitory neurons recorded with multielectrode arrays in the human temporal cortex during the wake-sleep cycle. Our analysis shows that the Ising model captures neuronal collective behavior much better than the independent model during wakefulness, light sleep, and deep sleep when both excitatory (E) and inhibitory (I) neurons are modeled; ignoring the inhibitory effects of I neurons dramatically overestimates synchrony among E neurons. Furthermore, information-theoretic measures reveal that the Ising model explains about 80-95% of the correlations, depending on sleep state and neuron type. Thermodynamic measures show signatures of criticality, although we take this with a grain of salt as it may be merely a reflection of long-range neural correlations.

Inferring interaction partners from protein sequences using mutual information.

Functional protein-protein interactions are crucial in most cellular processes. They enable multi-protein complexes to assemble and to remain stable, and they allow signal transduction in various pathways. Functional interactions between proteins result in coevolution between the interacting partners, and thus in correlations between their sequences. Pairwise maximum-entropy based models have enabled successful inference of pairs of amino-acid residues that are in contact in the three-dimensional structure of multi-protein complexes, starting from the correlations in the sequence data of known interaction partners. Recently, algorithms inspired by these methods have been developed to identify which proteins are functional interaction partners among the paralogous proteins of two families, starting from sequence data alone. Here, we demonstrate that a slightly higher performance for partner identification can be reached by an approximate maximization of the mutual information between the sequence alignments of the two protein families. Our mutual information-based method also provides signatures of the existence of interactions between protein families. These results stand in contrast with structure prediction of proteins and of multi-protein complexes from sequence data, where pairwise maximum-entropy based global statistical models substantially improve performance compared to mutual information. Our findings entail that the statistical dependences allowing interaction partner prediction from sequence data are not restricted to the residue pairs that are in direct contact at the interface between the partner proteins.