318

23

Regulatory Networks

small numberc much less than gc ≪g of control nodes, represented as ang times cg × c matrixupper RR. This implies

decomposition of the experimental g times tg × t matrix upper EE:

upper E equals upper R upper FE = RF

(23.4)

whereupper FF is ac times tc × t matrix giving the temporal evolution of the control nodes. However,

this decomposition is not, in general, unique. Inference of the network is still largely

a heuristic procedure, in which alternative topologies fitting the data equally well

are considered, and, finally, a selection is made on the basis of additional, ad hoc,

information.

Developments under way include Petri nets, which may be able to incorporate

more biological features while still retaining a compact description. Representing

network components as tensors allows many standard manipulations to be carried

out, some of which may turn out to be useful in revealing useful features of the data.

Explicit differential equations for regulation ¹⁷ may be useful for more complete

quantification. The temporal variation of expression of a gene product zz under the

effect of mm regulators is written as

StartFraction d z Over d t EndFraction equals StartFraction k 1 Over 1 plus exp left parenthesis minus sigma summation Underscript j equals 1 Overscript m Endscripts w Subscript j Baseline y Subscript j Baseline left parenthesis t right parenthesis plus b right parenthesis EndFraction minus k 2 z comma^dz

dt ⁼

k1

1 + exp(−^Σm

j=1 ^{w j y j(t) + b) −k2z ,}

(23.5)

where k 1k1 is the maximum rate of expression, the yy represent the expression levels

of the regulators (usefully approximated as polynomials) and ww are their rates, bb

represents delay, and k 2k2 is the rate coefficient for degradation of zz. This system of

equations can be fitted to experimental microarray data.

Another practically useful approach for genome-wide expression data, in which

the expression levels of upper GG genes are probed under upper MM experimental conditions, is to

transform it from genes times× microarrays space (represented by a upper G times upper MG × M matrix) to

a reduced “eigengenes” times× “eigenarrays” space, the latter being unique orthonormal

superpositions. ¹⁸ The transformation makes use of singular value decomposition

(SVD). ¹⁹ More power is brought to bear using tensor decomposition. ²⁰ In a similar

vein of integrating data from multiple sources (including gene ontology annotation)

is the hyperlink-induced topic search (HITS) algorithm further developed by Zhang

et al. (2020). Many tensor analysis applications in bioinformatics are derived from

the MultiAspectForensics tool for detecting and visualizing novel subgraph patterns

in networks introduced by Maruhshi et al. (2011). Contemporaneous work by Li et al.

should also be mentioned. One interesting feature of their work was the demonstration

of the value of weighting the network links (e.g., by dichotomizing with a threshold).

17 Vu and Vohradský (2007).

18 Alter et al. (2000).

19 Golub and Van Loan (2013), Sect. 2.4.

20 Hore et al. (2016).