BACKGROUND
ribonucleotide reductase  is a key component of de novo deoxynucleotide  metabolism and an important target of cancer therapies  <cit> .
this enzyme, which reduces ribonucleoside diphosphates into corresponding deoxyribonucleoside diphosphates, is exquisitely controlled to properly balance dntp fluxes in the face of changing scheduled  and unscheduled  dntp synthesis demands  <cit> .
recent data  <cit>  suggest that ribonucleotide reductase  exists not only as a heterodimer r12r <dig> of r <dig> and r <dig> homodimers  <cit> , but also as a r14r <dig> tetramer and as a r16r <dig> hexamer, where hexamer formation is driven by atp binding to a previously undescribed hexamerization site.
thus, in addition to its four substrate binding possibilities in adp, gdp, cdp, or udp, and four dimerization/specificity site binding possibilities in datp, atp, dgtp, or dttp, the current view  is that r <dig> has two tetramerization/inhibitory site binding possibilities in datp or atp, and one hexamerization/activation site binding possibility in atp, in addition to possibilities of unbound site states.
the resulting large number of possible r <dig> states implies an even larger number of quaternary states, and this leads to a complicated mathematical model of rnr activity  <cit> .
this model, although useful for explaining rnr activity data, is not useful for biochemical network simulations because: a) it is unwieldy , b) it uses different mathematical structures under different substrate-modulator conditions, and c) it requires root solutions of high order polynomials to determine r <dig> proportions in mono-, di-, tetra- and hexamer quaternary states, and thus rnr activity, as a function of modulator and total r <dig> concentrations.
simpler mathematical reaction rate models of rnr are needed if deoxynucleotide metabolism  <cit>  is to be represented using systems biology markup language   <cit> , a standard which requires single algebraic expression reaction rate laws in some applications  <cit> .
based on recent data from cooperman's group  <cit> , such expressions are provided here for rnr.
RESULTS
reaction activities are viewed here as weighted sums of enzyme state specific activities multiplied by probabilities of enzymes being in specific states.
for example, a michaelis-menten reaction rate law is viewed as
where the probability that the enzyme is in a loaded/reactive state  is p and the probability that the enzyme is in an empty/unreactive state  is p.
the rnr models presented here are based on the following four enzyme state probability assumptions:
<dig>  the probability that a particular r <dig> subunit is bound to ndp is assumed to be
<dig>  the probability that l ∈ {atp, datp, dttp, dgtp} is also bound to the dimerization/specificity site, conditional on ndp binding, is assumed to be
where, if l = dttp and ndp = gdp for example,
is the probability that dttp is bound to the dimerization/specificity site given that gdp is bound to the substrate site; the probability of an empty dimerization/specificity site is thus
<dig>  the probability that the tetramerization site is either empty, occupied by atp, or occupied by datp, is assumed to be, respectively,
<dig>  finally, the probability that the hexamerization site is occupied by atp is assumed to be
the subscripts a , i , and s  on the binding constants correspond to h , t , and d  subscripts on p, respectively.
that parameter values differ depending upon which ndp substrate is bound to the active site  is indicated by the conditional probabilities.
binding constants in μm, rate constants in 1/s.
afixed values taken directly from table  <dig> of  <cit> .
busing eq.
<dig>  these were adjusted to yield fluxes of  <dig> ,  <dig> ,  <dig> and  <dig>  for adp, gdp, cdp and udp, respectively, under assumptions of e <dig> =  <dig> μm, adp =  <dig> μm, gdp =  <dig> μm, cdp =  <dig> μm, udp =  <dig> μm, kmadp =  <dig> μm, kmgdp =  <dig>  μm, kmcdp =  <dig> μm, kmudp =  <dig>  μm, atp =  <dig> μm, datp =  <dig>  μm, dgtp =  <dig>  μm, dttp =  <dig> μm and the remaining parameter values in table  <dig>  see  <cit> .
cno data, thus, these can be assumed to have any value between .
<dig> and 2; a default value of  <dig> was carried down from gdp.
dno data,  <dig> is based on the other rows.
previous work  <cit>  has shown that the dimer and hexamer states are active, that the tetramer state is slightly active for adp and gdp and is otherwise inactive, that dimer state activity for cdp and udp exists when atp or datp is bound to the dimerization/specificity site, and that an empty dimerization/specificity site still permits the formation of some dimer with cdp reductase activity, see table  <dig> of  <cit> .
thus, based on the enzyme state probabilities given above, for kcat implicitly defined through
we propose the following expressions:
adp reduction
gdp reduction
cdp reduction
udp reduction
in these equations, for adp and gdp, the first factor is the probability that the dimer site is occupied, and for cdp and udp, the first factor is the expectation of kcat conditional on r <dig> being in a dimer state .
in the adp and gdp models, the second factor is the conditional expectation of kcat given that the dimerization site is occupied: the first term of this second factor has in its numerator the statement that kcat = k <dig> if the tetramerization site is empty, or kcat = k <dig> if it is occupied by either datp or atp, and the second term states that kcat = k <dig> if the hexamerization site is occupied by atp.
for the cdp and udp models, the first term of the second factor is the probability of an empty tetramerization site , and the second term states that if the atp concentration is high enough that the hexamerization/second term dominates the tetramerization/first term whilst the first factor approaches k2a, k <dig> is the overall kcat.
this rationale served as our model selection guide.
importantly, the models fitted recent data  <cit>  very well, see figure  <dig> and table  <dig>
discussion
in general, when an integrated system is engineered from component subsystems, the behavior of the overall system depends on component input-output specifications more so than the details of component implementations.
by analogy, when enzymological data are applied to biochemical network modeling, rather than the elucidation of reaction mechanisms, it can be expected that the reaction surfaces themselves  determine network behavior more so than the details of how such surfaces are represented.
thus, for applications to systems biology, large confidence intervals  in the model parameter estimates of table  <dig>  are not a problem because only goodness-of-fit  really matters; this claim assumes an operating range within the data range, since similarly fitting models often veer apart when used in extrapolations.
if reaction mechanism inferences were instead being sought, the large ci in the model parameter estimates would have been a problem, e.g.
the squared terms in the model suggest cooperative binding, but this choice provides only slightly better curve fits compared to linear terms , so cooperative binding cannot be inferred from this model.
in the rnr model presented here, the proportion of r <dig> units existing in monomer, dimer, tetramer, or hexamer states, and thus the rnr activity per unit enzyme, depends on site binding occupancies but does not depend on the total r <dig> concentration.
in the more complicated previous model  <cit> , higher total r <dig> concentrations favor higher order quaternary states.
the degree to which this is so is illustrated by plots of predicted gdp reductase activity as a function of atp concentration at various r <dig> concentrations .
consistent with the formation of higher order quaternary r <dig> states, these plots contract to the left as the total r <dig> concentrations increase from  <dig> μm to  <dig> μm.
in future work, the model given here will be altered to capture such trends without losing its simplicity; the total r <dig> concentration will enter such a model not only as a linear modulator of the reaction surface amplitude , but also as a modifier of reaction surface shape parameters, e.g.
kaatp will be replaced by a decreasing function of r <dig>
CONCLUSIONS
we identified a rational polynomial model of rnr activity that has single algebraic expressions for each reductase reaction rate law.
the expressions provide reasonably good fits  to recent data  <cit> .
compared to previous reaction rate expressions for this data  <cit> , the new expressions are simpler and thus better suited for biochemical network simulations, particularly those constrained to use enzyme reaction rate laws defined as single algebraic expressions  <cit> .

 ribonucleotide reductase is a key component of de novo deoxynucleotide metabolism and an important target of cancer therapies. this enzyme, which reduces ribonucleoside diphosphates into corresponding deoxyribonucleoside diphosphates, is exquisitely controlled to properly balance dntp fluxes in the face of changing scheduled and unscheduled dntp synthesis demands. recent data suggest that ribonucleotide reductase exists not only as a heterodimer r12r of r and r homodimers, but also as a r14r tetramer and as a r16r hexamer, where hexamer formation is driven by atp binding to a previously undescribed hexamerization site. thus, in addition to its four substrate binding possibilities in adp, gdp, cdp, or udp, and four dimerization/specificity site binding possibilities in datp, atp, dgtp, or dttp, the current view is that r has two tetramerization/inhibitory site binding possibilities in datp or atp, and one hexamerization/activation site binding possibility in atp, in addition to possibilities of unbound site states. the resulting large number of possible r states implies an even larger number of quaternary states, and this leads to a complicated mathematical model of rnr activity. this model, although useful for explaining rnr activity data, is not useful for biochemical network simulations because: a) it is unwieldy, b) it uses different mathematical structures under different substrate-modulator conditions, and c) it requires root solutions of high order polynomials to determine r proportions in mono-, di-, tetra- and hexamer quaternary states, and thus rnr activity, as a function of modulator and total r concentrations. simpler mathematical reaction rate models of rnr are needed if deoxynucleotide metabolism is to be represented using systems biology markup language, a standard which requires single algebraic expression reaction rate laws in some applications. based on recent data from cooperman's group, such expressions are provided here for rnr.  reaction activities are viewed here as weighted sums of enzyme state specific activities multiplied by probabilities of enzymes being in specific states. for example, a michaelis-menten reaction rate law is viewed as where the probability that the enzyme is in a loaded/reactive state is p and the probability that the enzyme is in an empty/unreactive state is p. the rnr models presented here are based on the following four enzyme state probability assumptions: the probability that a particular r subunit is bound to ndp is assumed to be the probability that l ∈ {atp, datp, dttp, dgtp} is also bound to the dimerization/specificity site, conditional on ndp binding, is assumed to be where, if l = dttp and ndp = gdp for example, is the probability that dttp is bound to the dimerization/specificity site given that gdp is bound to the substrate site; the probability of an empty dimerization/specificity site is thus the probability that the tetramerization site is either empty, occupied by atp, or occupied by datp, is assumed to be, respectively, finally, the probability that the hexamerization site is occupied by atp is assumed to be the subscripts a, i, and s on the binding constants correspond to h, t, and d subscripts on p, respectively. that parameter values differ depending upon which ndp substrate is bound to the active site is indicated by the conditional probabilities. binding constants in μm, rate constants in 1/s. afixed values taken directly from table of. busing eq. these were adjusted to yield fluxes of,, and for adp, gdp, cdp and udp, respectively, under assumptions of e = μm, adp = μm, gdp = μm, cdp = μm, udp = μm, kmadp = μm, kmgdp = μm, kmcdp = μm, kmudp = μm, atp = μm, datp = μm, dgtp = μm, dttp = μm and the remaining parameter values in table see. cno data, thus, these can be assumed to have any value between. and 2; a default value of was carried down from gdp. dno data, is based on the other rows. previous work has shown that the dimer and hexamer states are active, that the tetramer state is slightly active for adp and gdp and is otherwise inactive, that dimer state activity for cdp and udp exists when atp or datp is bound to the dimerization/specificity site, and that an empty dimerization/specificity site still permits the formation of some dimer with cdp reductase activity, see table of. thus, based on the enzyme state probabilities given above, for kcat implicitly defined through we propose the following expressions: adp reduction gdp reduction cdp reduction udp reduction in these equations, for adp and gdp, the first factor is the probability that the dimer site is occupied, and for cdp and udp, the first factor is the expectation of kcat conditional on r being in a dimer state. in the adp and gdp models, the second factor is the conditional expectation of kcat given that the dimerization site is occupied: the first term of this second factor has in its numerator the statement that kcat = k if the tetramerization site is empty, or kcat = k if it is occupied by either datp or atp, and the second term states that kcat = k if the hexamerization site is occupied by atp. for the cdp and udp models, the first term of the second factor is the probability of an empty tetramerization site, and the second term states that if the atp concentration is high enough that the hexamerization/second term dominates the tetramerization/first term whilst the first factor approaches k2a, k is the overall kcat. this rationale served as our model selection guide. importantly, the models fitted recent data very well, see figure and table discussion in general, when an integrated system is engineered from component subsystems, the behavior of the overall system depends on component input-output specifications more so than the details of component implementations. by analogy, when enzymological data are applied to biochemical network modeling, rather than the elucidation of reaction mechanisms, it can be expected that the reaction surfaces themselves determine network behavior more so than the details of how such surfaces are represented. thus, for applications to systems biology, large confidence intervals in the model parameter estimates of table are not a problem because only goodness-of-fit really matters; this claim assumes an operating range within the data range, since similarly fitting models often veer apart when used in extrapolations. if reaction mechanism inferences were instead being sought, the large ci in the model parameter estimates would have been a problem, e.g. the squared terms in the model suggest cooperative binding, but this choice provides only slightly better curve fits compared to linear terms, so cooperative binding cannot be inferred from this model. in the rnr model presented here, the proportion of r units existing in monomer, dimer, tetramer, or hexamer states, and thus the rnr activity per unit enzyme, depends on site binding occupancies but does not depend on the total r concentration. in the more complicated previous model, higher total r concentrations favor higher order quaternary states. the degree to which this is so is illustrated by plots of predicted gdp reductase activity as a function of atp concentration at various r concentrations. consistent with the formation of higher order quaternary r states, these plots contract to the left as the total r concentrations increase from μm to μm. in future work, the model given here will be altered to capture such trends without losing its simplicity; the total r concentration will enter such a model not only as a linear modulator of the reaction surface amplitude, but also as a modifier of reaction surface shape parameters, e.g. kaatp will be replaced by a decreasing function of r  we identified a rational polynomial model of rnr activity that has single algebraic expressions for each reductase reaction rate law. the expressions provide reasonably good fits to recent data. compared to previous reaction rate expressions for this data, the new expressions are simpler and thus better suited for biochemical network simulations, particularly those constrained to use enzyme reaction rate laws defined as single algebraic expressions. 
a mathematical model of rnr activity which explicitly represents the states of r currently exists, but it is complicated in several ways: it includes up to six-fold nested sums; it uses different mathematical structures under different substrate-modulator conditions; and it requires root solutions of high order polynomials to determine r proportions in mono-, di-, tetra- and hexamer states and thus rnr activity as a function of modulator and total r concentrations. we present four rational polynomial models of rnr activity as a function of substrate and reaction rate modifier concentrations. this large number of internal r states implies an even larger number of quaternary states. thus, the current view is that r has four ndp substrate binding possibilities, four dimerization site binding possibilities, two tetramerization site binding possibilities, and one hexamerization site binding possibility, in addition to possibilities of unbound site states. recent data suggest that ribonucleotide reductase exists not only as a heterodimer r12r of r and r homodimers, but also as tetramers r14r and hexamers r16r recent data also suggest that atp binds the r subunit at a previously undescribed hexamerization site, in addition to its binding to previously described dimerization and tetramerization sites. the new models avoid the complications of the earlier model without compromising curve fits to recent data. compared to the earlier model of recent data, the new rational polynomial models are simpler, adequately fitting, and likely better suited for biochemical network simulations.
 recent data suggest that ribonucleotide reductase exists not only as a heterodimer r12r of r and r homodimers, but also as tetramers r14r and hexamers r16r recent data also suggest that atp binds the r subunit at a previously undescribed hexamerization site, in addition to its binding to previously described dimerization and tetramerization sites. thus, the current view is that r has four ndp substrate binding possibilities, four dimerization site binding possibilities, two tetramerization site binding possibilities, and one hexamerization site binding possibility, in addition to possibilities of unbound site states. this large number of internal r states implies an even larger number of quaternary states. a mathematical model of rnr activity which explicitly represents the states of r currently exists, but it is complicated in several ways: it includes up to six-fold nested sums; it uses different mathematical structures under different substrate-modulator conditions; and it requires root solutions of high order polynomials to determine r proportions in mono-, di-, tetra- and hexamer states and thus rnr activity as a function of modulator and total r concentrations.  we present four rational polynomial models of rnr activity as a function of substrate and reaction rate modifier concentrations. the new models avoid the complications of the earlier model without compromising curve fits to recent data.  compared to the earlier model of recent data, the new rational polynomial models are simpler, adequately fitting, and likely better suited for biochemical network simulations. 