A previously described computational framework was used for simulating hypoxanthine deprivation [
15]. Briefly, alterations in transcriptomic data obtained under control and stressed conditions were used to simulate and describe the stress phenotype, and the approach was modified to include the effects of metabolite pool alterations, which may occur during the IDC under different conditions. Specifically, time-dependent metabolomic data obtained under hypoxanthine-rich and -deprived conditions were used as an additional constraints to the previous approach [
15].
To obtain the metabolic flux distribution of
P. falciparum during the IDC, the following equations were solved:
$${\text{Minimize}}\sum\limits_{j \in R} {\left| {v_{j} } \right|}$$
(1)
$${\text{subject to:}} \, S \cdot \overline{v} = 0$$
$$\sum\limits_{i \in M} {\left| {v_{i} - \widetilde{{m_{i} }} \cdot v_{i,nom} } \right|} < \delta$$
where
R denotes the set of all metabolic reactions of
P. falciparum,
M represents the set of metabolic reactions influenced by the metabolomic data,
S represents a matrix containing the stoichiometry coefficients of all reactions,
\(\overline{v}\) denotes a column vector containing all metabolic reactions of the network,
\(\delta\) denotes the minimum of
\(\mathop \sum _{i \in M} \left| {v_{i} - \widetilde{{m_{i} }} \cdot v_{i,nom} } \right|\) obtained separately,
vg represents the metabolic reaction governing the growth of the parasite,
\(\mu\) denotes the nominal value of the parasite growth rate, which is set to 0.48 g/h gDW of the original merozoite [
2], and
\(\widetilde{{m_{i} }}\) denotes the minimum value of
\(m_{i}\), where
\(m_{i}\) is a vector containing the median values of each metabolite taking part in the
ith metabolic reaction.
\(v_{i,nom}\) denotes the
ith value of
\(v_{nom}\), which is obtained by solving the following:
$$\hbox{min} \sum\limits_{j \in R} {\left| {v_{j} } \right|}$$
(2)
$${\text{subject to:}} \, v_{i} < v_{N} ,\quad \forall \, i \in N$$
here,
N denotes the set of reactions transporting nutrients across the parasite plasma membrane and
\(v_{N}\) denotes a vector containing the optimal values of every reaction in
N. The other variables are as defined above.
The optimization problems shown in Eqs. (
1) and (
2) yield
vref, which were modulated using the time-dependent transcriptomic and metabolomic data to obtain the temporal profile of
P. falciparum metabolism. The time-dependent transcriptomic and metabolomic data were incorporated by solving the following:
$$\hbox{min} \sum\limits_{j \in G;j \ne i} {\left| {v_{j}^{t} - r_{j}^{t} v_{j,ref} } \right|} + \sum\limits_{i \in G} {\left| {v_{i}^{t} - \overline{m}_{i}^{t} \cdot v_{i,ref} } \right|}$$
(3)
$${\text{subject to:}} \,S \cdot \overline{v}^{t} = 0$$
$$\sum\limits_{i \in G} {\left| {v_{i}^{t} - \overline{m}_{i}^{t} \cdot v_{i,ref} } \right|} \le \varepsilon$$
$$v_{k} < v_{N} ,\quad \forall \, k \in N.$$
here,
G denotes the set of all intracellular reactions of
P. falciparum,
\({\text{r}}_{\text{j}}^{\text{t}}\) denotes the reaction expression of the
jth reaction at time
t,
\(\overline{m}_{i}^{t}\) represents the influence of a metabolite at time
t for the
ith reaction, and the bar above
m denotes normalization of metabolite abundance by its median over the IDC.
\(v_{j,ref}\) and
\(v_{i,ref}\) represent the
jth and
ith values, respectively, of
\(v_{ref}\). As suggested by Fang et al. [
2], the problem shown in Eq. (
3) can have multiple solutions. Therefore, the following optimization problem was solved to obtain a solution closest to
\(v_{ref}\):
$$\hbox{min} \, \left\| {v^{t} - v_{ref} } \right\|$$
(4)
$${\text{subject to:}}\sum\limits_{j \in G;j \ne i} {\left| {v_{j}^{t} - r_{j}^{t} v_{j,ref} } \right|} + \sum\limits_{i \in G} {\left| {v_{i}^{t} - m_{i}^{t} \cdot v_{i,ref} } \right|} \le \delta$$
$$v_{i} < v_{N} ,\quad \forall \, i \in N$$
The method to solve the optimization problems in Eqs. (
2)–(
4) was identical to the method previously described [
15], except for the term incorporating the metabolomic data in Eq. (
3).