Isotopically Nonstationary MFA (INST-MFA) of Autotrophic Metabolism

Abstract

Metabolic flux analysis (MFA) is a powerful approach for quantifying plant central carbon metabolism based upon a combination of extracellular flux measurements and intracellular isotope labeling measurements. In this chapter, we present the method of isotopically nonstationary ¹³C MFA (INST-MFA), which is applicable to autotrophic systems that are at metabolic steady state but are sampled during the transient period prior to achieving isotopic steady state following the introduction of ¹³CO₂. We describe protocols for performing the necessary isotope labeling experiments, sample collection and quenching, nonaqueous fractionation and extraction of intracellular metabolites, and mass spectrometry (MS) analysis of metabolite labeling. We also outline the steps required to perform computational flux estimation using INST-MFA. By combining several recently developed experimental and computational techniques, INST-MFA provides an important new platform for mapping carbon fluxes that is especially applicable to autotrophic organisms, which are not amenable to steady-state ¹³C MFA experiments.

Appendix: Simple Network Example for INST-MFA Calculations

A simple metabolic network appears in Fig. 6 as an example of how to construct the stoichiometric matrix S, identify the set of EMUs required to simulate MIDs of measured metabolites, and set up dynamic isotopomer balances on these EMUs. Table 3 delineates the atom transitions for the network. In this network example, metabolite A is the sole substrate and metabolite G is the only final product. The intermediary metabolites B, C, D, E, and F are assumed to be at metabolic steady state but isotopically nonstationary.

Fig. 6

Simple metabolic network used to illustrate the decomposition into EMUs. Atom transitions for the reactions in this model are given in Table 3. The network fluxes are assumed to be constant since the system is at metabolic steady state. Extracellular metabolite A is assumed to be at a fixed state of isotopic labeling to which intracellular metabolites B, C, D, E, F, and G adapt over time

Table 3 Stoichiometry and atom transitions for the reactions in the example metabolic network

The stoichiometric matrix S is shown below, which has k = 5 intermediary metabolites and j = 8 fluxes, resulting in a 5 × 8 matrix:

$$ S=\left[\begin{array}{lllllllll} 1\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \end{array}\right] $$

Therefore, S∙v = 0 is expressed in vector form as

$$ \left[\begin{array}{ll} {v}_1-{v}_2-{v}_5+{v}_6\hfill \\{\hfill} {v}_2-{v}_3+{v}_4\hfill \\{\hfill} {v}_3-{v}_4+{v}_5-{v}_6-{v}_7\hfill \\{\hfill} -{v}_2+{v}_7\hfill \\{\hfill} {v}_3-{v}_4-{v}_8\hfill \end{array}\right]=0 $$

A systematic method of EMU network decoupling in which metabolite units are grouped into mutually dependent blocks is described through this simple network example. For this example, we will set up the simplest possible model to simulate the MID of metabolite C, i.e., EMU C₁₂₃. First, we need to identify all the possible EMUs that contribute to the formation of C₁₂₃—in this reaction model, C₁₂₃ is formed from the condensation of B₁₂ + E₁ and D₁₂ + F₁ in reactions 2 and 4, respectively. This is recorded and the process is then repeated for all new EMUs, starting with the largest EMU in size; in this case, all EMUs of size 3 have already been identified. Next, the process is repeated to determine all the EMUs of size 2 that were previously identified, starting with D₁₂. D₁₂ is formed from two different reactions—from B₁₂ in reaction 5 and from C₂₃ in reaction 3. Following this, we determine which reactions form C₂₃; C₂₃ is formed from D₁₂ in reaction 4 and B₂ + E₁ in reaction 2. Finally, we need to determine which reactions form B₁₂. B₁₂ is formed from A₁₂ and D₁₂ in reactions 1 and 6, respectively. A₁₂ is a network substrate and is not produced by any other reactions and D₁₂ has already been considered in the previous step. Therefore, all EMU reactions of size 2 have been identified. The process is repeated once again for EMUs of size 1, until all the EMUs have been traced back to network substrates or previously identified EMUs. Table 4 shows the complete EMU decomposition of this system, which involves 24 EMU reactions connecting 16 EMUs.

Table 4 Complete list of EMU reactions generated for metabolite C

After EMU decomposition, the reaction network can be further decoupled into blocks, which group together minimal sets of mutually dependent metabolite units that must be solved simultaneously. Figure 7 shows the EMU network decomposition for the simple network example after block decoupling. The blocks are arranged so that each one is a self-contained subproblem, which will depend on the outputs of the previously solved blocks. Therefore, EMUs in block 1 should first be solved, then block 2, etc.

Fig. 7

(a) EMU network decomposition for simple example network (Fig. 6) generated to simulate the labeling of metabolite C. The EMU network was decoupled based on EMU size and network connectivity. (b) EMU network decomposition for the same network using block decoupling. Subscripts refer to the atoms that are contained within the EMU

The EMU reactions obtained from network decomposition and block decoupling form the new basis for generating system equations. The decoupled blocks can be arranged into a cascaded system of ODEs with the following form, as described in Subheading 3.9:

$$ {\mathbf{ C}}_n\cdot \frac{\mathrm{ d}{\mathbf{ X}}_n}{\mathrm{ d}t}={\mathbf{ A}}_n\cdot {\mathbf{ X}}_n+{\mathbf{ B}}_n\cdot {\mathbf{ Y}}_n $$

The concentration matrix C _n is a diagonal matrix whose elements are pool sizes corresponding to EMUs represented in X _n . X _n comprises row vectors that represent the MIDs of each EMU and dX _n /dt is the time derivative of X _n . Analogously, the input matrix Y _n also comprises row vectors that represent MIDs of EMUs that have been previously calculated. The system matrices A _n and B _n come from calculating the “true” flux vectors (v) based on the chosen free fluxes (u) and null space matrix (N). Furthermore, in the decoupled blocks, the full MID of products formed from condensation reactions can be obtained from the convolution (or Cauchy product, denoted by “×”) of MIDs of preceding EMUs. In the case of C₁₂₃, these MIDs are B₁₂ and E₁ or D₁₂ and F₁, i.e., C₁₂₃ = B₁₂ × E₁ or C₁₂₃ = D₁₂ × F₁. The following equations represent the system of ODEs for the simple network example:

$$ \begin{array}{lll}\left[\begin{array}{lll}C_{\mathrm{ C}\hfill} & \hfill 0\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill {C}_{\mathrm{ B}\hfill} & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill 0\hfill & \hfill {C}_{\mathrm{ D}\hfill} \end{array}\right]\left[\begin{array}{lll}\displaystyle \frac{{\mathrm{ d}\mathrm{ C}}_2}{\mathrm{ d}t\hfill} \\{\hfill}\displaystyle \frac{{\mathrm{ d}\mathrm{ B}}_2}{\mathrm{ d}t\hfill} \\{\hfill}\displaystyle \frac{{\mathrm{ d}\mathrm{ D}}_2}{\mathrm{ d}t\hfill} \end{array}\right]=\left[\begin{array}{lllll} {-v}_2-{v}_4\hfill & \hfill {v}_2\hfill & \hfill {v}_4\hfill \\{\hfill} 0\hfill & \hfill -{v}_1-{v}_6\hfill & \hfill {v}_6\hfill \\{\hfill} {v}_3\hfill & \hfill {v}_5\hfill & \hfill -{v}_3-{v}_5\hfill \end{array}\right]\left[\begin{array}{lll} {\mathrm{ C}}_2\hfill \\{\hfill} {\mathrm{ B}}_2\hfill \\{\hfill} {\mathrm{ D}}_2\hfill \end{array}\right]\cr+\left[\begin{array}{lll} 0\hfill \\{\hfill} {v}_1\hfill \\{\hfill} 0\hfill \end{array}\right]\left[{\mathrm{ A}}_2\right]\end{array} $$

$$ \left[{C}_{\mathrm{ E}}\right]\left[\frac{{\mathrm{ d}\mathrm{ E}}_1}{\mathrm{ d}t}\right]=\left[-{v}_7\right]\left[{\mathrm{ E}}_1\right]-\left[{v}_7\right]\left[{\mathrm{ D}}_2\right] $$

$$ \begin{array}{lll}\left[\begin{array}{lll} {C}_{\mathrm{ C}\hfill} & \hfill 0\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill {C}_{\mathrm{ D}\hfill} & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill 0\hfill & \hfill {C}_{\mathrm{ B}\hfill} \end{array}\right]\left[\begin{array}{lll} \displaystyle\frac{{\mathrm{ d}\mathrm{ C}}_3}{\mathrm{ d}t\hfill} \\{\hfill}\displaystyle \frac{{\mathrm{ d}\mathrm{ D}}_1}{\mathrm{ d}t\hfill} \\{\hfill}\displaystyle \frac{{\mathrm{ d}\mathrm{ B}}_1}{\mathrm{ d}t\hfill} \end{array}\right]=\left[\begin{array}{lll} {v}_2+{v}_4\hfill & \hfill -{v}_4\hfill & \hfill 0\hfill \\{\hfill} -{v}_3\hfill & \hfill {v}_3+{v}_5\hfill & \hfill -{v}_5\hfill \\{\hfill} 0\hfill & \hfill -{v}_6\hfill & \hfill {v}_1+{v}_6\hfill \end{array}\right]\left[\begin{array}{lll} {\mathrm{ C}}_3\hfill \\{\hfill} {\mathrm{ D}}_1\hfill \\{\hfill} {\mathrm{ B}}_1\hfill \end{array}\right] +\left[\begin{array}{cc} {{v}_2} & 0 \cr{\hfill} 0\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill {v}_1\hfill \end{array}\right]\left[\begin{array}{ll} {\mathrm{ E}}_1\hfill \\{\hfill} {\mathrm{ A}}_1\hfill \end{array}\right]\end{array} $$

$$\begin{array}{cc} \left[\begin{array}{cc} {C}_{\mathrm{ C}\hfill} & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill {C}_{\mathrm{ F}\hfill} \end{array}\right]\left[\begin{array}{ll} \displaystyle\frac{{\mathrm{ d}\mathrm{ C}}_1}{\mathrm{ d}t\hfill} \\{\hfill}\displaystyle \frac{{\mathrm{ d}\mathrm{ F}}_1}{\mathrm{ d}t\hfill} \end{array}\right]=\left[\begin{array}{cc} -{v}_2-{v}_4\hfill & \hfill {v}_4\hfill \\{\hfill} {v}_3\hfill & \hfill -{v}_3\hfill \end{array}\right]\left[\begin{array}{ll} {\mathrm{ C}}_1\hfill \\{\hfill} {\mathrm{ F}}_1\hfill \end{array}\right]+\left[\begin{array}{ll} {v}_2\hfill \\{\hfill} 0\hfill \end{array}\right]\left[{\mathrm{ B}}_1\right] \end{array}$$

$$ \begin{array}{ccc}\left[\begin{array}{ccc} {C}_{\mathrm{ C}\hfill} & \hfill 0\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill {C}_{\mathrm{ D}\hfill} & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill 0\hfill & \hfill {C}_{\mathrm{ B}\hfill} \end{array}\right]\left[\begin{array}{ll} \frac{{\mathrm{ d}\mathrm{ C}}_{23}}{\mathrm{ d}t\hfill} \\{\hfill} \frac{{\mathrm{ d}\mathrm{ D}}_{12}}{\mathrm{ d}t\hfill} \\{\hfill} \frac{{\mathrm{ d}\mathrm{ B}}_{12}}{\mathrm{ d}t\hfill} \end{array}\right]=\left[\begin{array}{ccc} -{v}_2-{v}_4\hfill & \hfill {v}_4\hfill & \hfill 0\hfill \\{\hfill} {v}_3\hfill & \hfill -{v}_3-{v}_5\hfill & \hfill {v}_5\hfill \\{\hfill} 0\hfill & \hfill {v}_6\hfill & \hfill -{v}_1-{v}_6\hfill \end{array}\right]\left[\begin{array}{ll} {\mathrm{ C}}_{23\hfill} \\{\hfill} {\mathrm{ D}}_{12\hfill} \\{\hfill} {\mathrm{ B}}_{12\hfill} \end{array}\right]\cr+\left[\begin{array}{cc} {v}_2\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill 0\hfill \\{\hfill} 0\hfill & \hfill {v}_1\hfill \end{array}\right]\left[\begin{array}{ll} {\mathrm{ B}}_2\times {\mathrm{ E}}_1\hfill \\{\hfill} {\mathrm{ A}}_{12\hfill} \end{array}\right]\end{array} $$

$$ \left[{C}_{\mathrm{ C}}\right]\left[\frac{{\mathrm{ d}\mathrm{ C}}_{123}}{\mathrm{ d}t}\right]=\left[-{v}_2-{v}_4\right]\left[{\mathrm{ C}}_{123}\right]+\left[\begin{array}{cc} {v}_2\hfill & \hfill {v}_4\hfill \end{array}\right]\left[\begin{array}{ll} {\mathrm{ B}}_{12}\times {\mathrm{ E}}_1\hfill \\{\hfill} {\mathrm{ D}}_{12}\times {\mathrm{ F}}_1\hfill \end{array}\right] $$

Solving this system of ODEs will simulate the EMU labeling trajectories needed to calculate the time-dependent MID of metabolite C. The flux and pool size parameters can then be adjusted iteratively using an optimization search algorithm to converge on parameter values that minimize the lack of fit with experimental mass isotopomer data.