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Building Physical Network Models
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Integration of Single-Gene Knockout Expression Data
In pursuit of our aim of explaining only significant activation/repression responses of genes following a knockout, we derived potential functions for observations indexed by Κp - a set of significant knock-out effects in the available data at p-values < 0.02. The actual knock-out effects were hidden variables in the model, and their values can be associated with the noise in the data of knock-out experiments. Each pairwise effect was first tied to an unobserved variable that represents actual (as opposed to measured) knock-out effect. The actual knock-out effects were then associated with three types of variables along candidate pathways:
1. dei denoted the direction of a protein-protein interaction ei which specified the causal direction of how the interaction is used in a signal transduction cascade. For simplicity, we assumed here that each interaction has only one directional annotation, essentially which reduced the protein-protein interactions to (inferred) directed edges
2. sei denoted the sign of pairwise interaction ei representing activation (+) or inhibition (-)
3. kij denoted the effect of knocking out gene gi on gene gj. kij = -1 if gj is down-regulated, +1 if gj is up-regulated, and 0 if gj is unaffected by the knock-out
4. σij denoted the measurement of the knockout effect kij
5. πq denoted a pathway of physical interactions
The actual knock-out effect kij was tied to the measurement oij via a potential function Φij, analogous to the protein-DNA and protein-protein interaction data:
where the likelihood ratios were derived from the available error model. We explained each such knock-out effect with a cascade of molecular interactions. For example, if two genes gi and gj were of interactions connected via a path π, the path was directed from gi to gj, and the aggregate sign along π agreed with kij, then the path is said to explain kij. A valid path had to satisfy several additional constraints.
For π to qualify as an explanation for kij, it must satisfy the following conditions:
The end nodes of π were gi and gj
The last edge in π was a protein-DNA interaction
All the edges in π were in the forward direction (from gi to gj)
The signs of the edges along π were consistent with the sign of the knock-out effect
The length of π was less than a pre-defined upper bound
If intermediate genes along π have been knocked out, they also exhibited a knock-out effect on gj
A path which satisfied these conditions was a candidate explanation
for kij. We say that kij is explained by
the physical model if at least one path satisfied these conditions.
These conditions can be modified to incorporate simple notions of coordinate
regulation.
Conditions 1, 2, and 5 can be verified without knowing how
the edges are annotated, merely assuming that they exist (are present in
the set of measured interactions). Therefore, for each kij we identified
a set of connecting paths Пij={π1,…,πn}
which satisfied these conditions. Пij contained all
candidate paths which could in principle explain kij. A
candidate path, if selected, imposed the additional constraints on the variables of conditions 3 and 4, and that all the edges must exist along the path.
Let πa in Пij be a
candidate explanatory path of knockout effect kij. Let Ea denote the protein-DNA and protein-protein edges along πa. Let Xa, Sa, and Da denote the edge presence, signs, and directions along πa. Then πa
explains kij if the following conditions hold:
xe=1 for all xe in Xa.
de= de’ for
all protein-protein interaction directions de. de’
is pre-determined by our definition of path direction.
The potential function encoding these conditions was then expressed as:
where I(.) is a 0/1 indicator function.
When there were multiple candidate paths connecting gi
and gj, we required that the conditions along at
least one of the paths explain kij.
Encoding these OR-like constraints in a single potential
function was cumbersome, so instead we introduced auxiliary path selection variables and factored the potential function into terms corresponding to single paths. Recall that σija denoted the selection variable of path πa, σija=1
if πa was used to explain kij, and zero otherwise. Physically, σija represented whether
the pathway πa played a regulatory role in the context of
the specific experiment. The potential function ψija
was augmented with variable σija:
The potential function did not vanish even when the constraints were violated so as to allow other causes (those not included in the model) to explain the knock-out effects. Our experimental results were not sensitive to the specific value of ε1.
We constructed a potential function term ψijOR
to specify the condition that at least one candidate path was selected to
explain kij if kij was explained. Similar
to other potential functions, ψijOR
is a ``soft'' logical OR:
Combining the previous two equations, the potential function
associated with each pairwise knock-out effect is as follows: denote Xij,
Sij, Dij the edge presence, sign and direction
variables along all valid pathways connecting gi and gj; Xa, Sa, Da the variables along a specific path πa. Then
ψij returned a relatively high value if at least one path explained kij and selected paths satisfied all the conditions of explanation. Moreover, the returned value was higher if there were more paths which explained the knock-out effect. This bias encouraged parallel pathways as explanations (the effect was mediated by multiple alternative pathways).
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