1. Representation, Learning and Inference in Models of Cellular Networks
BMI/CS 576
Colin Dewey
Fall 2010

2. Various Subnetworks within Cells
metabolic: describe reactions through which enzymes convert substrates to products
regulatory (genetic): describe interactions that control expression of particular genes
signaling: describe interactions among proteins and (sometimes) small molecules that relay signals from outside the cell to the nucleus
note: these networks are linked together and the boundaries among them are not crisp

3. Figure from KEGG database
gene products
other molecules

4. Part of the E. coli Regulatory Network
Caption:
This figure illustrates the core of the GRN in E. coli, where TFs regulate other TFs [74]. Short horizontal lines from which bent arrows extend represent cis-regulatory elements responsible for the expression of the genes named below the line. When more than one TF regulates a gene, the order of their binding sites is as given in the figure. An arrowhead indicates activation and a horizontal bar indicates repression when the position of the binding site is known. If only the nature of TF regulation is known, without binding site information, ‘+’ and ‘−’ symbols indicate activation and repression respectively. These examples may be indirect rather than direct regulation. The circles with the different colours as given in the key represent the different families of DNA binding domains. The names of dominant regulators are in bold. FIS, factor for inversion stimulation; IHF, integration host factor. Modified with permission, from Madan Babu, M. and Teichmann, S. A., (2003), Nucleic Acids Res. 31, 1234–1244. © Oxford University Press.
Figure from Wei et al., Biochemical Journal 2004

5. A Signaling Network Figure from Sachs et al., Science 2005
Classic signaling network and points of intervention. This is a graphical illustration of the conventionally accepted signaling molecule interactions, the events measured, and the points of intervention by small-molecule inhibitors. Signaling nodes in color were measured directly. Signaling nodes in gray were not measured, but are presented to place the signaling nodes that were measured within contextual cellular pathways. The interventions classified as activators are colored green and inhibitors are colored red. Intervention site of action is indicated in the figure. Arcs are used to illustrate connections between signaling molecules; in some cases, the connections may be indirect and may involve specific phosphorylation sites of the signaling molecules (see Table 3 for details of these connections). This figure contains a synopsis of signaling in mammalian cells and is not representative of all cell types, with inositol signaling corelationships being particularly complex.
Figure from Sachs et al., Science 2005

6. Two Key Tasks
learning: given background knowledge and high-throughput data, try to infer the (partial) structure/parameters of a network
inference: given a (partial) network model, use it to predict an outcome of biological interest (e.g. will the cells grow faster in medium x or medium y?)
both of these are challenging tasks because typically
data are noisy
data are incomplete – characterize a limited range of conditions
important aspects of the system not measured – some unknown structure and/or parameters

7. Transcriptional Regulation Example: the lac Operon in E. coli
E. coli can use lactose as an energy source, but it prefers glucose.
How does it switch on its lactose-metabolizing genes?

8. The lac Operon: Repression by LacI
lactose absent  protein encoded by
lacI represses transcription of the lac operon

9. The lac Operon: Induction by LacI
lactose present  protein encoded by
lacI won’t bind to the operator (O) region

10. The lac Operon: Activation by Glucose
glucose absent  CAP protein promotes binding
by RNA polymerase; increases transcription

11. Network Model Representations
directed graphs
Boolean networks
differential equations
Bayesian networks and related graphical models
etc.

12. Probabilistic Model of lac Operon
suppose we represent the system by the following discrete variables
L (lactose) present, absent
G (glucose) present, absent
I (lacI) present, absent
C (CAP) present, absent
lacI-unbound true, false
CAP-bound true, false
Z (lacZ) high, low, absent
suppose (realistically) the system is not completely deterministic
the joint distribution of the variables could be specified by 26 × = parameters

13. Motivation for Bayesian Networks
Explicitly state (conditional) independencies between random variables
Provide a more compact model (fewer parameters)
Use directed graphs to specify model
Take advantage of graph algorithms/theory
Provide intuitive visualizations of models

14. A Bayesian Network for the lac System
Pr ( L ) Z L G C I
lacI-unbound
CAP-bound
absent
present
0.9
0.1
Pr ( lacI-unbound | L, I ) L I
true
false
absent
0.9
0.1
present
Pr ( Z | lacI-unbound, CAP-bound )
lacI-unbound
CAP-bound
absent
low
high
true
false
0.1
0.8

15. Bayesian Networks Also known as Directed Graphical Models
a BN is a Directed Acyclic Graph (DAG) in which
the nodes denote random variables
each node X has a conditional probability distribution (CPD) representing P(X | Parents(X) )
the intuitive meaning of an arc from X to Y is that X directly influences Y
formally: each variable X is independent of its non-descendants given its parents

16. Bayesian Networks
a BN provides a factored representation of the joint probability distribution Z L G C I
lacI-unbound
CAP-bound
x 2 = 20
this representation of the joint distribution can be specified with 20 parameters (vs. 191 for the unfactored representation)

17. Representing CPDs for Discrete Variables
CPDs can be represented using tables or trees
consider the following case with Boolean variables A, B, C, D
Pr( D | A, B, C ) A
Pr(D = T) = 0.9 F T B
Pr(D = T) = 0.5 C
Pr(D = T) = 0.8
Pr( D | A, B, C ) A B C T F
0.9
0.1
0.8
0.2
0.5

18. Representing CPDs for Continuous Variables
we can also model the distribution of continuous variables in Bayesian networks
one approach: linear Gaussian models U1 U2 … Uk X
X normally distributed around a mean that depends linearly on values of its parents ui

19. The Inference Task in Bayesian Networks
Given: values for some variables in the network (evidence), and a set of query variables
Do: compute the posterior distribution over the query variables
variables that are neither evidence variables nor query variables are hidden variables
the BN representation is flexible enough that any set can be the evidence variables and any set can be the query variables L I G C L G I C
lacI-unbound
CAP-bound Z
present ?
low
lacI-unbound
CAP-bound Z

20. The Parameter Learning Task
Given: a set of training instances, the graph structure of a BN
Do: infer the parameters of the CPDs
this is straightforward when there aren’t missing values, hidden variables L I G C L G I C
lacI-unbound
CAP-bound Z
present
true
false
low
absent
high
...
lacI-unbound
CAP-bound Z

21. The Structure Learning Task
Given: a set of training instances
Do: infer the graph structure (and perhaps the parameters of the CPDs too) L G I C
lacI-unbound
CAP-bound Z
present
true
false
low
absent
high
...