Modelling of biomolecular networks Bioinformatics Modelling of biomolecular networks UL, 2017, Juris Vīksna
Modelling of biological processes Regulation of transcription Protein interactions Metabolic interactions
Types of models for intracellular networks
Metabolic pathways
Metabolic pathways
Metabolic pathways
Signalling pathways
Gene regulatory networks
Gene regulatory networks
Gene regulatory networks
What GRN models are useful for? Simulation. For given initial conditions compute how the system evolves with time. Model checking. Does the model behave according to the specifications it was constructed? Reconstruction from data. From a given set of data from biological measurements construct a model tha explains that data. Constraints. For a given ‘stable behaviour’ find the minimal requirements on system parameters that must be satisfied to ensure that this behaviour takes place. System dynamics. Find all the possible ‘stable behaviours’ that the biological system might exhibit.
Boolean models of GNR
Boolean models of GNR Attractors and attractor basins Each attractor basin is supposed do describe a particular ‘stable behaviour’ – such as processes within cells of different types or the behaviour of a cell in different conditions.
Boolean models of GNR
Boolean models of GNR
Boolean models of GNR (2008) Most of theproposed models include some ‘non-biological’ parts that help to ensure that the model behaves according to specifications. Boolean networks do not allow to model continuous time.
Boolean model Drosophila development
Boolean model Drosophila development
Modelling with differential equations With well tuned parameters may work well for simulation of biological systems. Finding the "right" parameters, however, already isn't a simple problem.
Petri net models Petri nets were invented in August 1939 by Carl Adam Petri (at the age of 13 !!!) for the purpose of describing chemical processes. Simulation and "model checking".
Finite state linear models (FSLM) A.Brazma, T.Schlitt, 2005 can be regarded as extensions of Boolean models which also accounts for the changes of protein concentrations over time gene state is a Boolean function from the sates of binding sites protein concentrations changes linearly
Modelling of binding sites in simplest case two constants a>d, sites becomes active when concentration c exceeds d and inactive when c falls below d several binding factors can compete for one binding site, this also could influence the protein growth rate
Example of FSLM
Shapiro model of lambda phage (Science, 1995) A very detailed semi-formal model of -phage.
FSLM for lambda phage
FSLM for lambda phage
GNR models based on hybrid systems (2001) (2007) (2004) (2010) (2004) Advantages of hybrid system based models: combined discrete/continuous modelling seems well suited for GRNs there are well established mathematical techniques for analysis
A thermostat example of HS
HS – yet another example
On hybrid systems based models HS-based formalisms are becoming increasingly popular in latest few years. Adaptation of Temporal Evolution Model for Drosophila circadian cycle modelling (Fromentin et al, 2010).
HS model of Drosophila circadian cycle Constraints required for cyclical behaviour of the system:
Problems with HS based models? Usually the model behaviour is very dependent from quantitative values of parameters (concrete PL differential equation etc.) The formalism usually allows to model events that are of little or no biological importance (such as resetting variable values to zero), thus include features that makes analysis more difficult.
HSM – a restricted HS-based model
HSM – a restricted HS-based model H = M,X,T,F,MF M – a finite set of modes (1, 2, ...) X – a finite set of continuous variables (x1, x2, ...) T – a set of mode transitions (1 p 2, p = x c) F – a set of growth/degradation functions (f1, f2, ...) MF – mode-function assignments (MF: X F)
Restrictions imposed on HSM H = M,X,T,F,MF Transition (1 p 2) guards p has very simple form: x c or x c. Functions f are continuous and monotonous, i.e. either increasing or decreasing. The values of variables are governed only by functions from F, no discontinuous changes are allowed. A number of technical restrictions just to exclude ‘strange’ behaviours.
A simple HMS and its observational graph Observational sequence: Q = (), (), (), (), (), (), (), ()
A toy model of HSM
Frames of HSM HSM: H = M,X,T,F,MF Frame supporting HSM H: F = M’,X,T’,MF* The main distinction: The set of functions F is replaced by set {,,} and MF* only tells whether a particular variable x in a particular mode is growing, decreasing or staying constant. Assignment MF* must be consistent with MF. Sets M’ and T’ can be either the same as M and T, or their refinement.
Analogs of attractors for HSM (frames)? Since the state space for HSM can be any directed graph the only obvious analogue seems to be strongly connected components (SCC). We could define an ‘attractor’ as SCC without outgoing edges. However the whole state space often can be a single SCC... Can we somehow improve the analysis?
Frame refinements Changes of variable x in some mode : xc2 c1xc2
Frame refinements F F’ 1 2 Changes of variable x in some mode : xc2 c1xc2 F F’ xc1 xc1 t1 t3 t1 t3 1 t4 2 t2 t4 t2
Frame refinements and SCCs F F’ In this refined frame there is one more mode, however a single larger SCC has been split in two smaller ones. Thus for a given HSM we could look for a frame refinement having the ‘finest’ partition in SCCs.
Frame refinements and SCCs
Temporary and permanent SCCs For some SCCs it is possible to show (computationally) that the system will leave them within a finite amount of time. Such SCCs are called temporary. Non-temporary SCCs are called permanent.
State space of toy GRN
Finite state linear models (FSLM) (2003) Can be regarded as extension of Boolean model that also accounts for the changes of protein concentrations over time. Gene state is a Boolean function from the sates of binding sites. Protein concentrations change linearly.
Shapiro model of lambda phage (Science, 1995) A very detailed semi-formal model of -phage.
FSLM for lambda phage
Two stable behaviours of -phage
HSM for -phage Includes 11 genes: N, cI, cII, cIII, cro, xis, int, O, P, Q and an artificial gene Struc. 10 binding sites (6 of them can bind 2 different proteins). 11664 modes. 42 different orderings of binding affinities (number of these is hugely restricted by considering only those that are consistent with known biological facts). The analysis program constructer a refined HSM frame with 20000 modes and with only two attractors that correspond to known lysis and lysogeny behaviours!
Attractors describing lysis and lysogeny There are only two attractors in HMS lambda phage model and they correspond to lysis and lysogeny behaviours!
Lysis behaviour of -phage A one of the two attractors of -phage, this one describes its lysis behaviour.
Lambda phage model
Proposed mutations of -phage For ‘normal’ lysis behaviour to hold, we must have: 1) OL3 < OL2 < OL1 and OR3 < OR2 < OR1 (well known from biological experiments) 2) OR2 < OL2 (follows from results from our state space analysis) a),b),c) shows proposals for experiments that change the order for affinities for OR2 and OL2, and, according to model, should lead to an observationally different lysis-type behaviour.