Is Dynamic Bayesian Networks Apt To Predict Receptor Conformational States?

Part I. Introduction.


A motivation here is to predict specific conformational states of receptors in all possible multimeric complexes, and to approximate how the components may influence properties of one another in the complex.

Algorithms, which enable prediction of probable protomeric states in multimers, would define intracellular signalling cascades occurring individually, or simultaneously, via specific receptor activation or co-activation. By relating the information with other findings obtained for gene regulatory networks and intracellular protein interactions, full pictures of the intracellular cascades occurring in the cell would be gathered in detail, provided that the same or a very similar set of expression profile was obtained by microarray or RT-PCR in the studies to be compared.

Since information involved in the entire process of cell-signalling are hugely expansive, modelling the whole would yet be an over-ambitious project. A model which represents a local region for receptors could be made, ideally containing all relevant components of the environment; however, missing or excess components are inevitable, partly because few experimental procedures are completely free from a fractional possibility of false negatives or positives arising, though some may disagree on this notion. Model systems which fit into empirical evidences and hypotheses would be a theoretical evidence that could dissolve any inconsistencies observed in experimental findings, as well as allowing predictions for unobserved behaviours to be made.

Statistical inferences enable models to be constructed incorporating certain degrees of uncertainty. A brief description of Bayesian inference is given below.


Background

Bayesian Probability

Bayesian inference is simply expressed as

P(H|O) = P(O|H)P(H)/P(O)

where P(H) is prior probability over hypothesis, P(O) is marginal probability of observed, P(O|H) is conditional probability of observed for given hypothesis, and P(H|O) is posterior probability of hypothesis for given observation.
NB: The above O should not be confused with odd ratio. In expressing Bayes rule, E often denotes evidence observed; here, E denotes edges in the next section and so the duplicated meaning on a notation was avoided.

for more on Bayes theorem plato.Stanford.encyclopedia. Bayes-theorem


Bayesian Network

Bayesian network is a graphical model in which joint probability distribution of a sequence of random variables and conditional dependencies of the variables are presented in a directed acyclic graph (DAG).

DAG is formed by a collection of vertices (i.e. nodes) and directed edges (i.e. arrows) joined in a manner in which a sequence of edges initiated at a vertex never loop back to said vertex, hence it is an acyclic graph of directed sequences. Vertices at which edges are initiated are often called parents, which connect to children vertices.

A DAG is expressed as G = (V, E)

where V signifies a set of vertices of random variables {X1,X2,...,Xp}, and E denotes a set of edges between the variables (see FIG 1).



FIG 1. A simple Bayesian network

V = {1,2,3,4,5}, E = {(1,2), (1,3), (2,4), (3,4), (4,5)}, and
factorized joint probability: P(1,2,3,4,5) = P(1)P(2|1)P(3|1)P(4|3,2)P(5|4).





A Bayesian network can give answers to probabilistic queries about the variables contained and relevant interrelations. The network forecast the state of a sequence of variables when other variables of evidences are observed.


Machine Learning and Parameter Estimation

In statistical inferences, conditional probability often include unknown parameters which can be evaluated from data.

A statistical inference, the frequentist approach, has a method of maximum-likelihood estimation (MLE), wherein mean and variance were estimated from a given sample, then a unknown parameter value is determined in order for the observed outcome to be most probable. When the model depends on unobserved inferred variables, a parameter can be estimated by expectation-maximisation (EM) algorithm, performed iteratively in alternating two steps: 1) E step for expected likelihood of the inferred variables to be evaluated; and 2) M step that maximises the likelihood derived at E step to derive parameters (Dempster, Laird & Rubin 1977; Wu 1983).

In Bayesian approach, parameters are estimated by specifying a loss function, by selecting the value that minimises expected loss under the posterior probability distribution for loss function in order to avoid reaching a wrong estimate (reviewed by Brooks 2003).

Bayesian approach often employs Markov chain Monte Carlo (MCMC) method for faster approximation of complex data. A Markov chain is a string of observations generated for any value at each point in the string, to be dependent of the forward but not of the previous values. The algorithm behaves like an explorer who decides randomly where to go next from his current position; his slight preference for the higher ground allows peaks on the surface to emerge in the uncharted landscape. The random walker, MCMC explorer gathers a global scene fast, in contrast to a MLE explorer who focuses much at a high peak for a better resolution, tending to get stuck there (explained more with the same analogy by Brooks 2003).

A dynamic, Hidden Markov model (HMM) assess invisible unobserved states from the outcomes which depend on the states following posterior probability distribution. HMM could allow an infinite numbers of unknown states applying stochastic Dirichlet process as a probability distribution.

wikipedia.org/wiki/Hidden_Markov_model
JASSS Markov Chain Analysis

HMM has been implemented in programs for sequence analysis on motif search and homology search. An example is HMMER, an open source software for sequence analysis, written by Eddy SR in Howard Hughes Medical Institute.
HMMER.janelia.org


Bayesian Networks in Biology

Bayesian networks have been employed extensively in biology, notably in analysing gene regulatory networks (Perrin et al 2003; Husmeier 2003), protein interactions (Chen et al 2010; Xu et al 2011), signalling pathways (Bender et al 2010), phylogenic tree analysis (Rasmussen & Kellis 2011), natural selection (Geisler & Diehl 2002), and predicting secondary structures of proteins (Aydin et al 2011). These are just a few examples of numerous studies that applied Bayesian inference to give answers to various queries in biology.


The Challenge

Analysing heteromeric receptor association complexes and receptor-effector interactions in a given cell-type with dynamic Bayesian network

Once the expression profile of the membrane proteins and intracellular proteins that associate with GPCRs (G-protein subtypes, kinases, phosphatases, β-arrestins and so on) in a cell-type under a specified condition was obtained, then their interaction network could be modelled for the cell-type specific for the condition.

If relevant experimental observations have already been made in said cell-type under the same condition, for instances ligand binding profiles of receptors of interests, affinity of effectors to the receptors, phosphorylation profiles, the rate of internalisation, known oligomerisation tendencies, and known synergistic activity of the receptor with other receptors etc are available, these information would be in the model formation, or treated as hidden to estimate the validity of the model. Such models would likely include several unknowns including a dynamic oscillatory changes in intracellular calcium concentration upon receptor activation.

The model may indicate how each receptor’s function relates with the other receptor, as well as mapping probable interactions with other relevant components in the local condition.

A comparable set of models for particular receptors could be generated based on expression profiles of the cells under different conditions of the same cell-type, or on datasets of different cell-types that express the receptors of interests with slightly different component. The processes may reveal better detailed pictures as to how the receptors tend to operate and how receptors may adapt to local environmental changes.

Predicting protein interactions with dynamic Bayesian network has already been performed by many research groups. Some of these are going to be discussed in Part II, which follows this introduction.


Estimating conformational status of protomers in putative oligomeric complexes

A question here is: how plausible would it be to predict each conformational state of protomer in a putative multimeric complex in each specific condition: could that be estimated solely based on experimental observations?

Although unknown variables can be generated by models, there have to be a certain consistency in the patterns related to the receptor pharmacology.

It would be beneficial to define case by case whether a certain synergistic effect observed by experiments are due to intracellular signalling cross-talks by mutual components shared between two distinct pathways of independent receptor actions, or resulted from co-activation of two receptors that influence physically and directly the function of one another by associations.

The beauty of models implementing machine learning is that once a system is relatively established, it could be continuously developed forward to a realistic representation, with probable variations in hands in accordance with empirical findings obtained for different conditions. This venture should be worthy of some efforts.



References

Aydin Z et al 2011. BMC Bioinformatics. 12:154. doi:10.1186/1471-2105-12-154

Bender C et al 2010. Bioinformatics 26: i596–i602. doi:10.1093/bioinformatics/btq385

Brooks SP. 2003. Phil Trans R Soc Lond A. 361: 2681–2697. doi:10.1098/rsta.2003.1263.

Chen X et al 2010. Bioinformatics. 26: i334–i342. doi:10.1093/bioinformatics/btq175

Dempster AP, Laird NM & Rubin DB. 1977. J Roy Statist Soc Ser B. 39: 1–38.

Geisler & WS & Diehl EL. 2002. Phil Trans R Soc Lond B 357: 419-448. doi: 10.1098/rstb.2001.1055

Husmeier D 2003. Bioinformatics 19: 2271–2282. doi: 10.1093/bioinformatics/btg313

Perrin BE et al 2003. Bioinformatics 19: ii138–ii148. doi: 10.1093/bioinformatics/btg1071.

Rasmussen and Kellis 2011. Mol Biol Evol. 28: 273–290. doi:10.1093/molbev/msq189

Xu Y et al 2011. J. R. Soc. Interface. 8: 555–567. doi:10.1098/rsif.2010.0384

Wu CFJ. 1983. Ann. Statist. 1: 95–103.

On Network Modelling for Receptor Topology and Cooperativity in Simplicity and Receptor Allostery in Complexity


_______________________________________________________________________________________
The author’s note

This was written for a large part based on three review articles: two on network modelling by Agnati et al. (2003; 2007) and another on GPCR pharmacology by Kenakin and Miller (2010). An initial attempt was to highlight a few points in a compact form; the process was a little carried away with an attempt to merge concepts to reinvent a new. The purpose of this was not to summarise empirical evidences (as that’s been done in the reviews referenced), but was to encapsulate key points. Readers may notice a few traces of occasional speculative outbursts, which were opted to stay in this free form.
_______________________________________________________________________________________

Contents
0. A Network Modelling Method for GPCRs
1. Horizontal Molecular Network (HMN)
2. Vertical Molecular Network (VMN)
3. Allosterisms Affecting HMN and VMN
4. Modelling Discrete Dynamic Networks
________________________________________________________________________________________________

Specific Contents
Allostery: 1.3., 2.1., 2.3., 2.4., 3.
G-protein: 1.3., 2.1., 2.2., 3.4a., 3.4d., 3.4e., 4.5.
Oligomerisation: 1.2., 1.4., 1.5., 1.6., 2.3., 2.4., 2.5., 3.3b., 3.3.c., 3.4e., 4.5.
Protein Conformation: 1.1., 2II., 2III., 2.2., 2.3., 3.2b.
_______________________________________________________________________________________


0. A Network Modelling Method for GPCRs

A review for the network modelling and the idea behind: Agnati LF et al. (2003; 2007; 2009)

Method: Boolean network / elementary cellular automata (from random initial conditions)

Basic Concept:

0.1. A biological system can be perceived as a summation of networks comprising nodes connected by channels. The communications in the network arise via “one-to-one” wiring transmission (wT) and “one-to-many” volume transmission (vT).

0.2. A transmembrane receptor is an entity which is under the influence of chemical and physical forces from three different microenvironment: the extracellular fluid, phospholipid bilayer of the cell-membrane, and the intracellular cytosolic fluid. Local components in each microenvironment are expected to influence at varied degrees on the receptor conformation and thereby its function, as the receptor and local components communicate via wT or vT. Consequentially emerging two directional molecular networks can be considered: a horizontal molecular network (HMN) within the plane of the membrane, and a vertical molecular network (VMN) across the three microenvironment (Agnati et al. 2003; 2007).


1. Horizontal Molecular Network (HMN)

1.0. As transmembrane proteins in the amphipathic environment can exist in relatively ordered clusters, the communication networks potentially involve various intra- and inter-molecular interactions interweaved in a dynamic system.

1.1. The heterogeneity of protein components in the clusters makes the precise prediction of receptor conformation and function in a given cellular system effortful. Proteins can be morpheeinic, meaning that the proteins vary in conformation at different oligomerisation status and the proteins exist in alternative conformations, each being unique to each oligomeric assembly. Going beyond the original concept of homomeric morpheeins suggested by Jaffe (Jaffe 2005), receptors co-expressed and co-localised with other subtypes, different receptors and/or other proteins in the clusters could co-exist as a group of heteromeric morpheeins to a degree. A sum of such contributions at the cluster level may be sufficient to affect the vertical information flow (see 2.2, 3.3b). Taken that in mind, analysis of HMN is a requisite for predicting receptor conformation and function.

1.2. GPCRs have a propensity for forming homomeric or heteromeric dimers (reviewed by Pin et al. 2007; Milligan 2009). Over a decade ago, two plausible modes of GPCR dimerisation were postulated: domain-swapping and contact models. The domain swapping model considers a GPCR to be divisible into two units: one from the N-terminus to TM5, and another from TM6 to the C-terminus, with the third intracellular loop (ICL3) in between the two. Having TM5/TM6 as a dimer interface, the two units were suggested to be exchanged between receptors associating upon activation (Gouldson et al. 2000); this model is energetically less favourable than a simpler, contact model. In last few years, a number of crystal structures have been obtained for several GPCRs, including active conformations: an opsin with a C-terminal fragment of Gα subunit (Scheerer et al. 2008) and an agonist-bound A2A adenosine receptor (Xu et al. 2011); the experimental evidences suggest that seven TMs of a GPCR likely remain gathered in a bundle upon activation. A number of other evidences indicates that oligomerisation of GPCRs occur by contacting at various regions, notably at TM4, also TM5, TM1, as well as at the both terminal domains, and ICL3 etc. (reviewed by Milligan 2008; Fuxe et al. 2010).

1.3. Previously it was surmised that ligand-binding cooperativity observed experimentally may imply dimerisation. The idea was refuted by descriptions given to the observation for monomers, with focuses on G-protein catalytic kinetics and G-protein competition; hence concluded: “cooperativity is not due to an allosteric coupling of the two ligand-binding sites in a GPCR dimer, but rather to an allosteric coupling of the ligand-binding site of a monomeric receptor with the nucleotide site of the G protein” (Chabre, Deterre and Antonny 2009) (see also 3.4a).

1.4. For Family/class C GPCRs, oligomerisation is a prerequisite in receptor activation process (reviewed by Pin et al. 2005; 2009). For Family/class A GPCRs, oligomerisation could affect signal transduction and/or receptor internalisation for some receptors (reviewed by Agnati et al. 2003; Terrillon & Bouvier 2004a; Prinster, Hague & Hall 2005; Milligan 2008) (see 3.3c).

1.5. GPCR oligomerisation could affect cell-surface expression of the receptors by possible means of stabilising conformation by minimising free energy cost to solvent exposure during cellular transport. Oligomerisation favours extracellular domains associating with endoplasmic reticulum (ER)-resident chaperones; it could also cover ER-retention motifs present within intracellular domains (reviewed by Milligan 2010). This is a point to be considered in HMN regarding the proportion of receptors in a cluster population (see 4.5.).

1.6. Some GPCRs are known to form oligomers including other transmembrane proteins. The formation of such heteromers has key functional roles for certain GPCRs (e.g. CLR with RAMPs, Frizzled family receptors with specific co-receptors) (reviewed by Sexton et al. 2008; Schult 2010); how extensively such oligomerisations with non-GPCR can affect the function of many other GPCRs remains largely uncertain.


2. Vertical Molecular Network (VMN)

2.0. VMN concerns a flow of information from the extracellular space via cell membrane to the intracellular space: usually, extracellular ligands bind to receptors to alter conformational equilibrium, followed by specific signalling cascades in the cell. In many cases, a GPCR initiates the intracellular signals by activating an associated effector, a heterotrimeric G-protein.

2.1. A review by Kenakin and Miller (2010) provides detailed descriptions on the vertical information flow regarding receptor allostery, as well as horizontal contributions on the allostery (described in 3.). A brief summary of the conception of GPCR allostery is given below, applying the same terminology employed by Kenakin and Miller in the review above.

2I. Concept 1. An allosteric system has three major components:
modulator;
conduit; and
guest,
in the system information can flow bidirectionally from a modulator or from a guest: the role of the modulator and the guest is interchangeable (Tränkle et al., 1999).

2Ia. GPCR activation is described as an allosteric system with:
modulator (agonist)
conduit (receptor)
guest (G-protein, β-arrestin etc.)

There are other possible contributors to the receptor conformation including protein modifications (e.g. phosphorylation), presence of ions and other interacting proteins (i.e. second/third guests, and third-party guests etc.) (see 3.4e).

2II. Receptor conformation
A receptor dynamically alternates its conformation between low and high energy states. The probability of a receptor transitions to the alternative conformational state (e.g. from inactive to active) is a function of energy differences and the height of energy barrier between the two states. With a sufficient affinity to a receptor, a ligand alters the height of energy barrier and so the contour of the energy landscape (2III); G-protein coupling also similarly affects that. Some GPCRs remain at low energy states in the absence of agonist, whereas some displays constitutive activity on effector coupling in the absence of agonist (Kobilka & Deupi 2007).

2III. The landscape of free energy
GPCRs exist in ensembles of conformation induced by various forces including ionic, polar, and hydrophobic interactions; such forces occur within a polypeptide at two levels: among amino acid side-chains, and between the secondary structures. For the most probable distribution of the system concerned at equilibrium (i.e. Boltzmann distribution), lower energy states are preferred than disordered higher energy states: more stably folded conformations are found towards the bottom of energy well representing the local minimum free energy; and conformational entropy for the overall folding involving covalent bonds like disulfide contribute largely.

2.2. A GPCR endogenously has a number of active states which reflects a number of possible interactions between the receptor and any possible effectors including G-protein trimers in variation, β-arrestins and other scaffolding proteins. Ligands with a functional selectivity exploits this property of the receptor (see 3.4.).

2.3. Receptors are more likely modulated allosterically at less ordered higher energy states (Kenakin & Miller 2010).

2.4. This (2.3.) implies that the affinity of modulators to receptors could vary depending on the microenvironment at which energy states of the receptor are defined; certain ligands may display a lower affinity to certain oligomers in which the receptor is relatively more stabilised at low-energy conformations but that could readily be recognised by specific antagonists. A ligand may favour associating with a heteromer than a homomer and vice versa, accordingly to the thermodynamical differences existing in the oligomeric assemblies. Hence there emerges a variant of pharmacological profiles if oligomerisation is to cause a sufficient thermodynamical change on the protomer for the ligand of interests (see 3.3c.).

2.5. It is hypothesised that the level of constitutive activity may reflect how likely the receptor engages in stable oligomerisation orderly, based on the observations: 1.5 and 2.3.


3. Allosterisms Affecting HMN and VMN

3.1. In an environment dense with various molecular species, a simple system (2Ia.) would likely meet additional components. Kenakin and Miller (2010) categorised GPCR allosterisms into three kinds: classic guest allostery in which a guest ligand co-occupies a receptor with a modulator ligand; laterally directed allostery in which a guest protein interacts horizontally with a receptor; and cytosolic allostery with various possible cytosolic guests (i.e. 2Ia.).

3.2. Classic Guest Allostery

3.2a. A receptor can have two ligands bound simultaneously: a ligand is a major endogenous agonist referred to as an orthosteric ligand, and another recognises an alternative site is known as the allosteric modulator; the latter binding affects the effects on the receptor induced by the orthosteric ligand upon co-binding of the two. With the emergence of bitopic ligands which co-occupy the both sites (Valant et al., 2008; 2009; Antony et al., 2009), various ligands for the allosteric modulation of GPCRs are increasing in numbers, each kind prospectively having unique pharmacological profiles (reviewed by Keov, Sexton & Christopoulos 2010).

3.2b. Here is a description on the actions of the allosteric modulators as the alteration of energy landscape: an allosteric modulator can produce either a positive or a negative effect by altering the energetic state of the receptor towards an ordered low-energy state by a negative allosteric modulator (NAM), or a more volatile higher-energy state by a positive allosteric modulator (PAM). Some allosteric modulators may apparently not belong to neither category; this may be the case when the modulator binds to a receptor in a manner stabilising one site on the expense of disordering the other; hence displaying a mixed characteristics.

3.3. Lateral Allostery

3.3a. The lateral allostery concerns horizontal protein-protein interactions within a phospholipid bilayer of a cell-membrane. The bilayer contains lipid species, notably cholesterols, cholesterol-binding proteins such as caveolins, and also some membrane proteins with sterol-sensing domains that recognises cholesterols; at the C-terminal domain GPCRs can be palmitoylated; these lipid components and the interacting proteins could affect conformation of a GPCR, its localisation, and its pharmacology (reviewed by Ostrom & Insel 2004; Chini & Parenti 2009) as well as of other transmembrane receptors (reviewed by Insel & Patel 2009).

3.3b. In an allosteric system involving a heteromeric receptor dimer, there are two possibilities: 1) the dimer can be a conduit, and a ligand binding to a protomer affects the ligand binding property of the other protomer (Rocheville et al. 2000; Marcellino et al. 2008) or the signalling property of the other (Hilairet et al. 2003; Ferré et al. 2007); and 2) a conduit receptor can form a dimer with another receptor which behaves as a modulator that affects the pharmacology of the conduit receptor (AbdAlla et al. 2000; El-Asmar et al. 2005; Ellis et al. 2006; Rashid et al. 2007).

3.3c. Experimental evidences suggest that heterodimerisation can affect receptor pharmacology variably among certain heteromers. Such effects include alterations in: ligand binding, G-protein-coupling, G-protein activation, and β-arrestin association (reviewed by Fuxe et al. 2008; Milligan 2009; Rozenfeld & Devi 2011).

3.4. Cytosolic Allostery

3.4a. In many cases, the major guest (see 2Ia) to GPCRs at the cytosolic side is a range of heteromeric G-proteins. G-protein subunits have various subtypes, each with specific effectors, and potentially produces diverse effects on the cell as well as initiating key signalling cascades via Gα-subunits (reviewed by Milligan & Kostenis 2006). A GPCR associates with the C-terminal portion of a Gα-subunit (Ślusarz & Ciarkowski 2004; Kostenis et al. 2005). Most receptors can associate with more than one type of Gα with own tendency to preferences that in some cases overlapping: for instance, receptors which couple to Gα12 can also bind to Gαq (Riobo & Manning 2005). The fine differences in the conformation of receptors associating with different Gα are recognisable by functionally selective ligands (Fisher et al. 1993; Gurwitz et al. 1994; Zhang, Brass & Manning 2008).

3.4b. The other notable guest is β-arrestins, largely composed of β-strands. As well as with a high affinity to phosphorylated GPCRs, β-arrestins interact with various proteins in signalling pathways, linking a GPCR-signal to another signal initiated by other receptors. Also by interacting with proteins which participate in protein trafficking, β-arrestins facilitate endocytosis of the receptors (reviewed by Claing et al. 2002; Gurevich & Gurevich 2006). Agonist-occupancy is not a necessity for β-arrestin-binding to a receptor, since β-arrestins can associate with certain receptors without an agonist occupying (Terrillon & Bouvier 2004b). Some agonists selectively bind to β-arrestin-bound receptors and produce effects inducible by β-arrestins: these are known as biased agonists (reviewed by Rajagopal, Rajagopal & Lefkowitz 2010).

3.4c. The phosphorylation of GPCRs, for enabling β-arrestins binding, is catalysed by G-protein receptor kinases (GRKs) (reviewed by Kohout & Lefkowitz 2002). Selective ligands could also be designed to recognise receptors phosphorylated by a specific GRK subtype among others, for specific GRK-GPCR association could determine the β-arrestin function (Zidar et al. 2009).

3.4d. There is a spatial limitation in receptors accommodating multiple guests. Gαq-like Gα15, which displayed higher affinities for certain receptors than Gαq or Gαs, was shown to decrease β-arrestin-induced desensitisation of the receptors (Innamorati et al. 2009).

3.4e. As GPCRs oligomerise, the capacity of receptors to accommodate cytosolic guests might vary among possible oligomers in a given cell system: if receptor A, which prefers only the company of a particular G-protein, heterodimerises with receptor B that likes inviting a certain GRK and β-arrestin as well, then homomer A, homomer B and heteromer A/B might display different pharmacology. It could also be plausible, regarding receptors like morpheeins, that heteromer A/B might agree to invite a novel guest which they would not invite otherwise as homomers, or disagree on certain guests. Furthermore, some guests might accompany one or more third-party guests, who bring more participants downstream. For some proteins keenly interacting with many others, the assembly could end up as a large loud party, which inevitably gets neighbours’ attention and responses - negative or positive. The type of response may depend on the economic situation of the cell: in a proliferating cell, the neighbours might be uplifting and cheerfully join the party; but in a poor condition, the neighbours could be miserable and annoyed, seeking any mean to shut the party down.


4. Modelling Discrete Dynamic Networks

4.1. According to Agnati et al., biological systems are composed based on three reasons: mosaic, a self-similarity logic, and biological attraction principle:
“Self-Similarity Logic indicates the self-consistency by which elements of a living system interact, irrespective of the spatio-temporal level under consideration. The term Mosaic indicates how, from the same set of elements assembled according to different patterns, it is possible to arrive at completely different constructions: hence, each system becomes endowed with different emergent properties. The Biological Attraction principle states that there is an inherent drive for association and merging of compatible elements at all levels of biological complexity.” (Agnati et al. 2009).

4.2. Boolean models for regulatory networks can replicate the emergence of self-assembly formation by iterative process. In the initially random Boolean networks, each network state is attracted to an attracter known as the basin of attraction, to which transient states flow approaching a periodic attractor limit cycle as branching patterns emerge (Wuensche 1994). It is possible to raise a complexity from a simple binary state machine, as initially demonstrated by Turing (1937).

4.3. A cluster of oligomers including various type of GPCRs within the plasma membrane was termed receptor mosaic by Agnati and Fuxe (reviewed by Fuxe et al. 2008). In the previous study by Agnati et al., receptor mosaics were modelled as a cluster of N nodes of a receptor with two hypothetical domains as defined in domain-swapping model (see 1.2.), in analysing receptor topology and cooperativity (Agnati et al 2007).

4.4. In the study 4.3., two receptor states: inactive (‘0’) and active (‘1’) were considered as a binary variable, and the receptor can be in two possible states: S = {1, 0}. The state of each receptor (i) is influenced by interactions which the receptor establishes, hence concerned are three inputs: the number and the state of neighbouring receptors; ligand binding; intracellular processes such as phosphorylation to favour ‘0’. The receptor states evolve in time through a sequence of discrete steps; at each step, the three inputs above define the state of each node. A receptor state (Si) at the time step (t +1) emerges as

Si (t + 1) = f ( Σ JijSi(t) - Ei(t) + Li(t)) = f (x)
when f(x) = 1 (x > 0), f(x) = 0 (x ≤ 0).

where Jij signifies coefficient for the coupling strength of the node i and j ; Ei is the effect of phosphorylation on the node i (hence negative); and Li denotes the effect of ligand on the node i.

NB: Jii ≠ 0 as a system default;
also Jij > 0 (∀i, j) for cooperativity (Koshland & Hamadani 2002).

4.5. In the system 4.4., a state-space of the receptor mosaic can be defined as the set of all probable configurations: the number of probable receptor mosaic configuration is 2 (power of N) for N number of receptors. In a finite state-space, a succession of configurations eventually gets trapped in a cycle of repetitive configurations, the attractor; the attractor period could be once in a fixed, or could be enormous in a chaotic system. Eventually the field self-organises itself into a limited number of attractors to reveal how the mosaic may behave (Agnati et al 2007).

4.6. At least two variants of the method 4.4. could be proposed: instead of the two hypothetical domains, a functional heterodimer, or alternatively a specific receptor-G-protein pair can be considered.

4.7. Bornholdt proposed an extended version of Boolean network with a touch of stochasticity:
by adding a concentration dynamics to a node, the input from neighbours are summed and either grow or decay with concentration variable ci(t) at each node; an explicit time delay td to count the transmission time of the incoming signals is introduced, and td is allowed to fluctuate for noise; then different processes, for the negative sum is a decay, or for the positive sum is a growth, results: that can be described by a simple differential equation driven by a binary input with a defined threshold for each node (Bornholdt 2008).

4.8. The extended network 4.7. presents the node which are not synchronously updated in discrete time steps as each node obeys its own autonomous dynamics; but the original synchronised dynamics can be restored by turning the noise down. This could be useful in eliminating attractors which are artefacts of the synchoronous clocking, as these likely disappear in the presence of noise (Klemn & Bornholdt 2005).

4.9. Here are tips from Bornholdt (2008):
“The simple steps to apply this technique are the following: (i) identify interaction network—make sure you have full knowledge of the network. Where unsure, make several variants of the network. (ii) Translate into a switching network. (iii) Simulate. (iv) Compare with known dynamical sequence data. Is not this what we do in our minds when drawing signalling networks on the blackboard?”



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Systems Modelling Tool

Allosteric Network Compiler (ANC)

Allosteric_Network_Compiler

by Ollivier JF, Shahrezaei V. and Swain PS

Scalable rule-based modelling of allosteric proteins and biochemical networks. PLoS Comput Biol. 2010 Nov 4;6(11):e1000975.

Method: Rule-based, with integrated free energy-based constraints

Efficiency: Combinatorial complexity is minimised by employing a few (2 or 3) intensive parameters.

Characteristics:

The rule-based system built on Monod-Wyman-Changeux paradigm of allostery of two states (T:tense or R:relaxed), the system employs: a regulatory factor gamma (Γ) which gives fold-changes of equilibrium constant; and one or more phi-values which give the effect of the ligands on forward and reverse rates in allosteric transition.

The model comprises a set of modular structures and interaction rules.

Each structure has a set of named components. There are two types of components: 1) hierarchical components that can either be compositing/constituting higher structures (domains, subunits) or as allosteric it undergoes conformational transition according to two-states model; and 2) the three interaction sites (catalytic, covalent-modification, ligand-binding).

Rules specify the interactions between the sites, and also how a conformational state or a modification state of the protein affects the strength of the associations. Rule thereby gives dissociation rates according to the nature of interactions.

Considering an allosteric protein as a modular, a binding modifier is an input for the output: the lengths of time at which the receptor component spent in each conformation when the allosteric transition is at equilibrium.

Free energy-based constraints is integrated as thermodynamic framework, to determine the collective effects of multiple modifiers on the conformational equilibrium. All modifiers are regarded to interact independently with each conformational state of the protein body, contribute independently to the free energy, and affects kinetic rates independently.

After a series of iterations, a biochemical reaction network is obtained; that is

exportable by Facile to perform deterministic or stochastic simulation in maths softwares such as MATLAB and Mathematica.

For GPCRs

Cubic ternary complex model is implemented with a single allosteric component and receptor with two conformational states, regardless of cooperativity in ligand binding, because introducing cooperativity factors into the model increases combinatory complexity. For a better efficiency, a sequential allosteric model termed quartic ternary complex model was introduced. The quartic model comprises two coupled allosteric components: an extracellular domain (ED) which binds to a ligand existing in transition between low and high-affinity states (s and t), and an intracellular domain (ID) which associates with G-protein and in transition between active and inactive states (i and a). The two components are linked, and each domain can be a modifier to the other by associations which are parameterised by the factor Γ and phi values. No parameter in the model is for a ligand--G-protein pairing.

The quartic model has 11 parameters including 3 intensive ones.

The quartic model is modular and is expandable to incorporate additional interactions such as oligomerisation.

However, the performance may deteriorate as the compiled network increase in size.