Automated mass action model space generation and analysis methods for two-reactant combinatorially complex equilibriums: An analysis of ATP-induced ribonucleotide reductase R1 hexamerization data

Biology Direct BioMed Central Research Open Access Automated mass action model space generation and analysis methods for two-reactant combinatorially complex equilibriums: An analysis of ATP-induced ribonucleotide reductase R1 hexamerization data Tomas Radivoyevitch Address: Department of Epidemiology and Biostatistics, Case Western Reserve University, Cleveland, Ohio 44106, USA E-mail: Published: 9 December 2009 Biology Direct 2009, 4:50 doi: 10.1186/1745-6150-4-50 Received: 2 December 2009 Accepted: 9 December 2009 This article is available from: http://www.biology-direct.com/content/4/1/50 © 2009 Radivoyevitch; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Background: Ribonucleotide reductase is the main control point of dNTP production. It has two subunits, R1, and R2 or p53R2. R1 has 5 possible catalytic site states (empty or filled with 1 of 4 NDPs), 5 possible s-site states (empty or filled with ATP, dATP, dTTP or dGTP), 3 possible a-site states (empty or filled with ATP or dATP), perhaps two possible h-site states (empty or filled with ATP), and all of this is folded into an R1 monomer-dimer-tetramer-hexamer equilibrium where R1 j-mers can be bound by variable numbers of R2 or p53R2 dimers. Trillions of RNR complexes are possible as a result. The problem is to determine which are needed in models to explain available data. This problem is intractable for 10 reactants, but it can be solved for 2 and is here for R1 and ATP. Results: Thousands of ATP-induced R1 hexamerization models with up to three (s, a and h) ATP binding sites per R1 subunit were automatically generated via hypotheses that complete dissociation constants are infinite and/or that binary dissociation constants are equal. To limit the model space size, it was assumed that s-sites are always filled in oligomers and never filled in monomers, and to interpret model terms it was assumed that a-sites fill before h-sites. The models were fitted to published dynamic light scattering data. As the lowest Akaike Information Criterion (AIC) of the 3-parameter models was greater than the lowest of the 2-parameter models, only models with up to 3 parameters were fitted. Models with sums of squared errors less than twice the minimum were then partitioned into two groups: those that contained no occupied h-site terms (508 models) and those that contained at least one (1580 models). Normalized AIC densities of these two groups of models differed significantly in favor of models that did not include an h-site term (Kolmogorov-Smirnov p < 1 × 10-15); consistent with this, 28 of the top 30 models (ranked by AICs) did not include an h-site term and 28/30 > 508/2088 with p < 2 × 10-15. Finally, 99 of the 2088 models did not have any terms with ATP/R1 ratios >1.5, but of the top 30, there were 14 such models (14/30 > 99/2088 with p < 3 × 10-16), i.e. the existence of R1 hexamers with >3 a-sites occupied by ATP is also not supported by this dataset. Conclusion: The analysis presented suggests that three a-sites may not be occupied by ATP in R1 hexamers under the conditions of the data analyzed. If a-sites fill before h-sites, this implies that the dataset analyzed can be explained without the existence of an h-site. Page 1 of 19 (page number not for citation purposes) Biology Direct 2009, 4:50 http://www.biology-direct.com/content/4/1/50 Reviewers: This article was reviewed by Ossama Kashlan (nominated by Philip Hahnfeldt), Bin Hu (nominated by William Hlavacek) and Rainer Sachs. Background Introduction The dNTP supply system produces dNTPs at rates that match those demanded by DNA replication and repair. With respect to ribose ring moieties, it is comprised of both a de novo system whose substrates are ribonucleoside diphosphates (NDPs) and a salvage system whose substrates are deoxynucleosides (dNs). The de novo system includes ribonucleotide reductase (RNR), deoxycytidylate deaminase (DCTD), and thymidylate synthetase (TS), and the salvage system includes deoxycytidine kinase (dCK), thymidine kinase 1 (TK1), deoxyguanosine kinase (dGK) and thymidine kinase 2 (TK2), see Fig. 1. The dNTP supply system is important because many anticancer agents target or traverse it (e.g. gemcitabine, hydroxyurea, triapine, 5-FU) or damage DNA directly (e.g. ionizing radiation, alkylating agents, oxaliplatin) and thus place demands on it for replacement dNTPs. In the future, mathematical models of cancer relevant systems will be needed to optimize multi-agent anticancer dose timings [1]. For example, gemcitabine (dFdC, diflourodeoxycytdine) [2] absorption is rate limited by dCK [3,4], dFdC targets RNR [5], dFdC resistance is associated with RNR over expression [6,7], and differential ionizing radiation (IR) sensitivity that dFdC imparts onto mismatch repair (MMR) defective cells may be due to mismatches caused by dNTP pool imbalances caused by RNR inhibition, rather than differential dFdC incorporation into DNA [8], so mathematical models of dNTP supply will be needed to optimize dFdCIR therapies of MMR defective cancers; MMR defective Figure 1 The dNTP supply system. Thick lines are fluxes, thin solid lines are activations, thin dashed lines are inhibitions. Key enzymes are described in the text. An RNR s-site mediated large positive feedback loop ATP Æ dCTP Æ dUMP Æ dTTP Æ dGTP Æ dATP terminates when dATP binds the R1 a-site to inhibit all 4 RNR reductions. Models of enzymes of this system will eventually be useful in anticancer drug dose time course optimizations [1]. Page 2 of 19 (page number not for citation purposes) Biology Direct 2009, 4:50 cancers are significant as they comprise ~10% of colorectal [9], gastric [10], pancreatic [11], urinary [12], gynecologic [13,14] and glioma [15] cancers. The dNTP supply system is ideal for cancer systems biology research because, among cancer relevant processes, it is perhaps the best understood. This is important because, intuitively, the more understanding a mathematical model captures, the more likely it is to be more useful than a conceptual model. Thus, the dNTP supply system is well poised to be successfully controlled better with mathematical modeling than without, and because of this, this system could become a standard of success in systems biology; the basis of this argument is prior success in the use of mathematical models to improve the control of well understood systems such as power plants and airplanes. RNR (NDP Æ dNDP) [16] has two subunits, R1, and R2 or p53R2 [17,18]. On short time scales of seconds to minutes, RNR is controlled through two R1 regulatory sites, a selectivity (s-) site that is somewhat analogous to a radio tuning control knob, and an activity (a-) site that can be thought (...truncated)