1. Modelling of biomolecular networks
Bioinformatics
Modelling of biomolecular networks
UL, 2017, Juris Vīksna

2. Modelling of biological processes
Regulation of transcription
Protein interactions
Metabolic interactions

3. Types of models for intracellular networks

4. Metabolic pathways

5. Metabolic pathways

6. Metabolic pathways

7. Signalling pathways

8. Gene regulatory networks

9. Gene regulatory networks

10. Gene regulatory networks

11. 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.

12. Boolean models of GNR

13. 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.

14. Boolean models of GNR

15. Boolean models of GNR

16. 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.

17. Boolean model Drosophila development

18. Boolean model Drosophila development

19. 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.

20. 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".

21. 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

22. 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

23. Example of FSLM

24. Shapiro model of lambda phage
(Science, 1995)
A very detailed semi-formal model of -phage.

25. FSLM for lambda phage

26. FSLM for lambda phage

27. 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

28. A thermostat example of HS

29. HS – yet another example

30. 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).

31. HS model of Drosophila circadian cycle
Constraints required for cyclical
behaviour of the system:

32. 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.

33. HSM – a restricted HS-based model

34. 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)

35. 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.

36. A simple HMS and its observational graph
Observational sequence:
Q =
(), (), (), (),
(), (), (), ()

37. A toy model of HSM

38. 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.

39. 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?

40. Frame refinements  Changes of variable x in some mode : xc2 c1xc2

41. Frame refinements F  F’ 1  2 Changes of variable x in some mode :
xc2
c1xc2
F  F’
xc1
xc1 t1 t3 t1 t3 1  t4 2 t2 t4 t2

42. 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.

43. Frame refinements and SCCs

44. 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.

45. State space of toy GRN

46. 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.

47. Shapiro model of lambda phage
(Science, 1995)
A very detailed semi-formal model of -phage.

48. FSLM for lambda phage

49. Two stable behaviours of -phage

50. 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!

51. Attractors describing lysis and lysogeny
There are only two attractors in HMS lambda phage model and they correspond to lysis and lysogeny behaviours!

52. Lysis behaviour of -phage
A one of the two attractors of -phage,
this one describes
its lysis behaviour.

53. Lambda phage model

54. 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.