Quantifying biochemical reaction rates from static population variability within incompletely observed complex networks

Abstract

Quantifying biochemical reaction rates within complex cellular processes remains a key challenge of systems biology even as high-throughput single-cell data have become available to characterize snapshots of population variability. That is because complex systems with stochastic and non-linear interactions are difficult to analyze when not all components can be observed simultaneously and systems cannot be followed over time. Instead of using descriptive statistical models, we show that incompletely specified mechanistic models can be used to translate qualitative knowledge of interactions into reaction rate functions from covariability data between pairs of components. This promises to turn a globally intractable problem into a sequence of solvable inference problems to quantify complex interaction networks from incomplete snapshots of their stochastic fluctuations.

Determining Interaction Directionality in Complex Biochemical Networks from Stationary Measurements

“…Furthermore, we have found a connection between the index of dispersion D and the ability to infer the direction of the interaction, where larger D yields a larger success ratio. Both degradation times and index of dispersion effect on the overall performance were previously demonstrated to be with similar conclusions as is given in [42]. Moreover, some of the network topological features such as the node degree were also being inspected.…”

“…The stationary Master equation yields the global balance relation, which means that for every node i where and the angular brackets represent conditional means. The above relation is derived from a summation of the Master equation over all other variables where consider stationarity ∂ t P = 0 [39–42], see further derivation in Methods and SI.…”

“…How-ever, while Eq. (1) must provably hold for all stationary states, empirically observed distributions carry sampling errors that can significantly impact inferred rates [42], leading to potential errors in determining the interaction direction. Our results reflect this errors dependence, as illustrated in Fig.…”

A mathematically rigorous algorithm to define, compute and assess relevance of the probable dissociation constant for every reaction of a constrained biochemical network

“…𝒮 𝓅 𝑧 ⊂ ℤ 𝐽×𝐼 ̿ ≝ 𝑆𝑡𝑜𝑖𝑐ℎ𝑖𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑛𝑢𝑚𝑏𝑒𝑟 𝑚𝑎𝑡𝑟𝑖𝑥 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑒𝑑 𝑏𝑖𝑜𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑛𝑒𝑡𝑤𝑜𝑟𝑘 (24) 𝑁(𝒮 𝓅 𝑧 ) ≝ 𝑁𝑢𝑙𝑙 𝑠𝑝𝑎𝑐𝑒 𝑜𝑓 𝒮 𝓅 𝑧 (25) 𝒱 ≔ 𝑆𝑢𝑏𝑠𝑝𝑎𝑐𝑒 𝑜𝑓 𝑁(𝒮 𝓅 𝑧 ) 𝑤𝑖𝑡ℎ 𝑐𝑎𝑟𝑑𝑖𝑛𝑎𝑙𝑖𝑡𝑦 #𝒱 𝐷𝑒𝑓. (15) 𝑘 = 1,2 … #𝒱 (26) Although the comprehensive subspace defined for the null space of the stoichiometric number matrix of the steady-state of a biochemical network is theoretically sound, the summations, for the 𝑢 𝑡ℎ -iteration will result in vectors that are trivial and redundant.…”

“…The availability of genome-scale data has permitted investigators to analyse biochemical networks for several organisms and from single cells with varying time points [22][23][24].…”