(125d) Model Misspecifications in Metabolic Flux Analysis: Biases, Tests and Fixes

Model
Misspecifications in Metabolic Flux Analysis: Biases, Tests and Fixes

Rudiyanto Gunawan*^,** , Sandro
Hutter*^,**

*Institute for
Chemical and Bioengineering, ETH Zurich, Zurich, Switzerland

**Swiss Institute of Bioinformatics, Lausanne,
Switzerland

Metabolic flux analysis (MFA) has become an indispensible tool
in metabolic engineering, not only for analyzing metabolic phenotypes (flux
distribution), but also for optimizing metabolic pathways to improve
productivity and/or to gain new capabilities. The MFA is based on a
stoichiometric model of the metabolic reaction network under the pseudo-steady
state assumption. The simplest variant of the MFA, hereupon referred to as the overdetermined MFA, uses a reduced stoichiometric model of
the cellÕs metabolism such that the flux estimation constitutes an over- (or
fully) determined linear regression problem.  Thanks to its simple formulation and
numerical implementation, the overdetermined MFA is still
widely used in metabolic engineering applications [1-2].

Despite the long history of MFA, the impact of
misspecifications of the stoichiometric model, particularly the issue of
missing or omitting reactions, has not received much attention. A recent study
demonstrated that modeling errors could lead to gross uncertainty in the
estimated intracellular flux values [3]. In this study, we employed the
framework of linear least square regressions to quantify the effects of missing
reactions on the flux estimates, and evaluated several statistical tests for
detecting omitted reactions in the stoichiometric model. Finally, we proposed
an iterative algorithm for resolving the issue of missing reactions in the overdetermined MFA.

The MFA can be formulated as a linear least square
regression problem as follow:

where S denotes
the stoichiometric matrix, v denotes
the fluxes, the subscripts I and E refer to the intracellular and
exchange (uptake) fluxes, respectively, and e denotes the vector of measurement errors. In the overdetermined MFA, the matrix S_Ihas a full column rank and the exchange fluxes v_E
are measured (or could be computed from the measurements). Assuming that the
error e has a zero mean and a constant
variance, the ordinary least square (OLS) solution provide the minimum variance
unbiased estimates (MVUE) for the intracellular fluxes v_I as follow:

When the variance of the error is not constant but known,
one can use the generalized least square estimation to provide the MVUE for v_I. When
the true model is in fact given by:

one can show that the OLS solution may no longer the MVUE
because

has a possible specification bias of

In the above derivation, the subscript O refers to the omitted reactions. Thus, the specification bias is
non-zero when the omitted fluxes v_O are non-zero and the missing reaction
stoichiometry is not orthogonal to S_I.

We demonstrated using a Chinese hamster ovary (CHO)
metabolic model and cell culture data that the omission of even a single
reaction from the network could lead to disproportionally large specification
biases in the intracellular flux estimation. Importantly, while a poor
statistical significance of the linear regression is a good indication of large
specification biases, an acceptable statistical significance can still result
from the omission of a reaction that causes significant flux biases.

We further evaluated three tests for detecting missing
reactions, including RamseyÕs RESET test, F-test for model selection, and the
Lagrange multiplier (LM) test [4]. Using 1000 randomly generated biochemical
reaction networks [5] and in silico flux values, we computed the true positive,
false positive, true negative and false negative rates of the above tests. The
results showed that the F-test robustly outperforms the RESET and LM tests across
different network sizes, numbers of omitted reactions, numbers of exchange
fluxes and levels of errors.

Based on the above findings, we then proposed an iterative
strategy to resolve the issue of missing reactions using the F-test. Briefly,
in each iteration, we employed the F-test to detect whether the addition of a
candidate reaction would lead to a statistically significant improvement in the
linear regression. We then incorporated the reactions that passed the
statistical threshold into the model, and repeated the F-tests for the
remaining reactions against the updated model. Using the CHO metabolic model
above and in silico
generated data, we demonstrated that the proposed iterative strategy could
recover the majority of missing reactions (~80%), and leave out reactions that
were not part of the reaction network.

The findings of this study are available in a pre-print
publication [6].

References:

1. Naderi, S.;
Meshram, M.; Wei, C.; Mcconkey, B.; Ingalls, B.; Budman, H.; Scharer, J.
Development of a mathematical model for evaluating the dynamics of normal and
apoptotic Chinese hamster ovary cells. Biotechnol. Prog. 2011, 27,
1197–1205.

2. Nolan, R. P.;
Lee, K. Dynamic model of CHO cell metabolism. Metab. Eng. 2011, 13,
108–24.

3. Sokolenko, S.; Quattrociocchi, M.; Aucoin, M. G.
Identifying model error in metabolic flux analysis – a generalized least
squares approach. BMC Syst. Biol. 2016, 10, 1–14.

4. Long, J. S.; Trivedi, P. K. Some Specification
Tests for the Linear Regression Model. Sociol. Methods Res. 1992,
21, 161–204.

5. Aho, T.; Smolander, O.-P.; Niemi, J.; Yli-Harja,
O. RMBNToolbox: random models for biochemical networks. BMC Syst. Biol. 2007,
1, 22.

6. R. Gunawan; S. Hutter. Assessing and Resolving
Model Misspecifications in Metabolic Flux Analysis. Preprints. 2017,
doi:10.20944/preprints201703.0124.v1.