Mechanism-Based Mutual Information between Metabolic Pathways Captures Pathway-Specific
Responses to Nutritional and Temperature Stresses. Shivani Patel1, Karl Carmona2 and Sungchul Ji1,
1
Department of Pharmacology and Toxicology, Ernest Mario School of Pharmacy, and 2Department of
Cell Biology and Neuroscience, School of Arts and Sciences, Rutgers University, Piscataway, N.J.
sji@rci.rutgers.edu.
Using DNA microarrays, Garcia-Martinez et al. [1] and Castells-Roca et al. [2] measured both transcript
levels (TLs) and transcription rates (TRs) from budding yeast cells at 6 time points upon applying
nutritional stress (switching the energy source from glucose to galactose) or temperature stress (raising
temperature of growth medium from 25 to 37° C). When the time-dependent TR and TL values of an
RNA molecule are plotted, a phase diagram is obtained consisting of 5 segments or vectors, each
spanning one of the 5 time intervals defined by the 6 time points of measurement. The angles of these
vectors are calculated and used to infer the mechanisms (of which there are 9) of interaction between
the transcription and transcript degradation steps that determines the RNA metabolic kinetics encoded
by a given gene. If a typical metabolic pathway consists of, say, 20 RNA molecules, there will be 20x5 =
100 mechanism-indicating numbers ranging from 1 to 9. By counting the number of times each of these
numbers appears in the phase diagrams belonging to a given pathway, it is possible to construct a
histogram for that pathway consisting of 9 columns whose height representing the probability of the
occurrence of (or activating) one of the 9 mechanisms in the pathway during an experiment. These
histograms can be used to calculate Shannon entropy, H, and mutual information, MI, based on the
following formulas: H(A) = -Sum (pi(A)log2 Pi(A)), where pi(A) is the probability of the ith mechanism
being activated in Pathway A, and MI(A,B) = Sum (pi(A,B) log2[(pi(A, B)/pi(A) x pi(b)], where pi(A, B) is the
joint probability of the ith mechanism being activated in both Pathways A and B, and pi(A) and Pi(B) are
marginal probabilities. The index i runs from 1 to 9. We studied the Shannon entropies of 10 pathways
and the mutual information values between all possible pairs among the 10 pathways. Each pathway
generates one H value and 9 MI values and the average of the latter has been interpreted as reflecting
the average tendency of a given pathway to couple with other pathways through one or more of the 9
possible mechanisms. The H and MI values were measured before replacing the original set of RNA
molecules belonging to a pathway with an equally sized set of randomly chosen RNAs in order to discern
if the H and MI values are pathway-specific. The H and MI values calculated thus are presented in the
following in the format, Metabolic pathway (Hbefore → Hafter under nutritional stress; MIbefore →MIafter
under nutritional stress; Hbefore → Hafter under temperature stress; MIbefore →MIafter under temperature
stress): ATP synthesis (2.42 → 2.63; 2.41→2.5; 1.46 → 2.11; 1.94 →1.97); DNA repair (2.28 → 2.51; 2.38
→2.64; 1.72 → 2.10; 1.80 → 1.87); Fatty acid metabolism (2.64 → 2.68; 2.59 → 2.47; 1.87 → 2.10; 1.86
→ 1.94); Glycolysis (2.43 →2.72; 2.42→2.37; 1.65 → 2.10; 1.90 → 1.77); Mating (2.43 → 2.35; 2.71 →
2.57; 1.59 →1.96; 1.77 → 1.95); Meiosis (2.41 → 2.56; 2.51→ 2.60; 1.82 → 2.13; 1.91→ 1.98);
Mitochondrial protein targeting (2.46 →2.47; 2.54 → 2.79; 1.74 → 2.01; 2.00 → 1.91); Oxidative
phosphorylation (2.53 →2.56; 2.57 → 2.52; 1.55 → 1.95; 1.92 → 1.82); Phospholipid metabolism (2.50 →
2.40; 2.54 → 2.50; 1.79 → 1.94; 1.97 → 1.93); Tricarboxylic acid cycle (2.43 → 2.54; 2.49 → 2.43; 1.62 →
2.11; 1.90 → 1.92). A statistical analysis of these value indicate (i) that Shannon entropies of the 10
pathways are similar with values around H = 2.5 and are unaffected by stresses nor by randomization of
RNAs and (ii) that the MI values of all of the 10 the pathways increase upon randomization under the
nutritional stress, and iii) that only about 40% of the 10 pathways show increased MI values upon
temperature stress.