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# The Biomass Objective Function

^{a}GT Life Sciences, Inc., 10520 Wateridge Circle, San Diego, CA, 92122, United States

^{b}Department of Bioengineering, University of California, 9500 Gilman Drive #0412, San Diego, La Jolla, CA 92093-0412, United States

## Abstract / Summary

Flux balance analysis (FBA) is a mathematical approach for analyzing the flow of metabolites through a metabolic network. To computationally predict cell growth using FBA, one has to determine the biomass objective function that describes the rate at which all of the biomass precursors are made in the correct proportions. Here we review fundamental issues associated with its formulation and use to compute optimal growth states.

## Introduction

Flux balance analysis (FBA) [1] is a widely used approach for studying biochemical networks, in particular the genome-scale metabolic network reconstructions that have been built in the past decade [2,3]. These network reconstructions contain all of the known metabolic reactions in an organism and the genes that encode each enzyme. FBA calculates the flow of metabolites through this metabolic network, thereby making it possible to predict the growth rate of an organism or the rate of production of a biotechnologically important metabolite. An objective function, such as the biomass objective function, is necessary to compute an optimal network state and resulting flux distribution (unique or non-unique) in a constraint-based reconstruction as the solution space is often very large for genome-scale networks [4]. With metabolic models becoming available for a growing number organisms [5] and high-throughput technologies enabling the construction of many more each year [6], FBA is an important tool for harnessing the knowledge encoded in these models.

Genome-scale models are used to compute a variety of phenotypic states. How the genome-scale metabolic network supports the growth of a cell has been a topic of much interest. Here we, 1) discuss the computation of cellular yields and growth rates and how they differ, 2) outline the formulation of a detailed biomass objective function, and 3) review several studies that have focused on the use of the objective function.

## Computing Cellular Yields and Growth Rates

Metabolic network reconstructions contain the known biochemical conversions inside the cell and allow for computation of both topological properties and biophysical capabilities. The vast majority of cellular metabolic conversions are enzymatically catalyzed with a few occurring spontaneously. A curated metabolic reconstruction can be utilized as a comprehensive parts list of the cell, allowing for detailed and accurate computation of the conversion of substrates into products by the cell.

### Computation of yields

Metabolic reconstructions are an ideal platform for rapidly calculating the yield of any given product from single or multiple substrates. Most often, the product yield (the maximum amount of product that can be generated per unit of substrate, Y_{p/s}) is of greatest interest (Figure 1). Calculation of biomass yields are different in that multiple biomass components (e.g., lipids) and biomass precursors (e.g., amino acids) have to be quantified in proportion to each other to form a biomass objective function (Figure 1). By detailing the molar content that makes up the biomass of the cell, stoichiometrically based biomass yields can be computed. The yield does not have a time dimension

### Computation of growth rate

Optimal or sub-optimal actual growth rates can also be computed. The growth rate is constrained by the measured substrate uptake rates, with the uptake rate of the limiting substrate being critical, and by maintenance energy requirements (Figure 1). Simulating the generation of cellular biomass products from available inputs using the biomass objective function allows for the prediction of allowable growth rates for given substrate uptake rates and maintenance requirements. The non-growth associated maintenance and the substrate uptake rate introduce time and thus enable the computation of a growth rate.

## The Formulation the Biomass Objective Function

The formulation of a detailed biomass objective function for use in examining metabolic networks is dependent on knowing the composition of the cell and energetic requirements necessary to generate biomass content from metabolic precursors (Figure 2). One can formulate biomass objective function at a different level of detail.

### Basic level

The formulation process starts with defining the macromolecular content on the cell (i.e., weight fraction of protein, RNA, lipid, etc.) and then the metabolites that make up each macromolecular group (e.g., amino acids, nucleotide triphosphates, etc.). With this information, it is possible to detail the required amount of metabolites (subsequently defining amounts of carbon, nitrogen, and additional elemental requirements) that are needed along with associated reaction pathways.

### Intermediate level

It is possible to increase this level of resolution and calculate the necessary biosynthetic energy that is needed to synthesis the macromolecules whose building blocks are directly accounted for in a curated metabolic network. For example, it is known that it takes approximately 2 ATP molecules and 2 GTP molecules to drive the polymerization of each amino acid into a protein molecule [7]. More energy is required when considering processes such as RNA error checking in transcription. This energetic conversion is included in the biomass objective function and details the necessary energy that the cell has to make to drive these biosynthetic processes (often included as part of maintenance energies). This energy is, of course, over and above the energy that is necessary to synthesize the appropriate macromolecular building blocks (e.g., the amount of energy to make a building block, such as UTP, from a common substrate, such as glucose). An important detail to take into account in the biomass objective function is that it is necessary to include the products of macromolecular biosynthesis from building blocks included in a network (e.g., water from protein synthesis and diphosphate from RNA or DNA synthesis). These polymerization products are then directly available to the cell and reduce the amounts of resources the cell needs to take up from the media.

### Advanced level

Advanced biomass objective functions can be formed by detailing the necessary vitamins, elements, and cofactors required for growth as well as determining core components necessary for cellular viability. Inclusion of vitamins, elements, and cofactors allow for the analysis of a broader coverage of network functionality and required network activity. Another advanced approach is to not only define the wild-type biomass content of the cell, but to generate a separate biomass objective function that contains the minimally functional content of the cell. This objective function (referred to as the ‘core’ biomass objective function [8]) can result in increased accuracy when predicting gene, reaction, and metabolite essentiality and is formulated using experimental data from genetic mutants and knockout strains. Workflows for how a biomass objective function is formulated have appeared [5,9]. Furthermore, a detailed spreadsheet of actual data used for formulating both a wild-type and core biomass objective function is available for *E. coli* [8] that can be used as a template for similar organisms.

The scope of network reconstructions continues to grow [5]. It should be noted that with full reconstructions of the entire protein synthesis machinery [10], that the level and detail in biomass objective functions can continue to grow.

## Brief Review of Studies Examining Cellular Objective Functions

Over the past two decades, an number of studies have been carried out to examine the use of objective function optimization with reconstructed networks towards predicting biological outcomes (Table 1) [11-19]. These studies have utilized small-scale central metabolic networks, as well as genome-scale reconstructions of bacteria and eukaryotic organisms. This set of studies can roughly be divided into two categories: (1) studies examining hypotheses on presumed cellular objective functions through comparison to experimental data [11-13,15,16,19], and (2) studies examining optimization techniques to discover or algorithmically predict biological objective functions from experimental data [14,17,18]. Each category is described below.

### Biased search for cellular objectives

Several studies have been conducted to examine which hypothesized cellular objective function best predicts cellular behavior through network optimization and comparison to experimental data. The first of these highlighted studies to appear (conducted in two parts [11,12]), considered growth of a hybridoma cell line with the intention of examining growth limiting substrate conditions and intracellular energy generation and utilization. This study utilized the wealth of information available for a known hybridoma cell line [20] to reconstruct its cellular network and investigate growth capabilities apparent from the stoichiometry of the network.

Later, studies in this category examined a number of additional cellular objectives to analyze growth characteristics of microorganisms and a growth-rate dependent biomass objective function [13]. One particular study performed a relatively comprehensive analysis of eleven different objective functions and compared each to growth of *E. coli* under six different growth conditions (the study also examined a number of different modeling parameters and their effect on phenotype prediction [16]). This combinatorial engineering approach of analyzing each objective functions towards predicting each experimental condition resulted in the findings that growth under batch (unlimited) and chemostat (limited) conditions are best described by two different cellular objectives. Another study in this category examined the metabolic burden of plasmid-based expression in a cell and has implications in biotechnology applications [19].

Examining the conclusions of each of these studies (see Table 1), two main points emerge: (1) the search for cellular objective functions is an ongoing area of research, and (2) objective functions for an organism are likely condition-dependent and training-data (comparison data) specific. Therefore, it is likely necessary to analyze the use of an objective function on a case-by-case basis for an intended application and useful to compare predicted fluxes to numerous input, output, and intracellular training-data fluxes in order to find the best overall predictive objective function.

### Unbiased search for cellular objectives

Studies of metabolism have also been conducted which utilize computational algorithms to determine best-fit cellular objective functions [14,17,18]. The details of each algorithm will not be discussed her, but these optimization-based frameworks each approach the determination of a predictive objective function in a different manner, and can also be utilized as tools to improve reconstructed network content [18]. In contrast to the studies where objective functions are first identified and then tested (described above), two effectively unbiased studies where an objective function was not initially assumed, concluded that optimization of biomass production or growth is the best fit for predicting growth data in the microorganisms *E. coli* [14] and *S. cerevisiae* [17]. The third study in this category developed an algorithm to refine both reconstruction and biomass objective function content, and demonstrated that overall improvements in cellular phenotype predictions can be achieved in such an approach (e.g., an increase in growth phenotype prediction of mutants from 91.4% to 96.7% in E. coli [18]). These algorithmic tools are readily applicable towards additional organism-specific networks and should aid in discovery projects, as well as industrially relevant applied applications.

## Conclusions

The biomass objective function describes the growth requirements of a cell. It is needed to perform a variety of Constraint-Based Reconstruction and Analysis (COBRA) methods [21]. It has a variety of uses ranging from the interpretation of evolutionary outcomes [22-24] to the introduction of a plasmid into a cell through the creation of additional metabolic burden [19]. Its use can allow for the computation of fluxes and provide insights into the functioning of cellular processes [25].

What does a microorganism try to do in a given environment? The answer to this question may be unknowable without understanding the evolutionary history of the target organism. Thus, we have a fundamental question associated with the selection of an appropriate objective function that is physiologically realistic. This issue was recognized in the very first paper on large scale network analysis using FBA [11,12] where a series of selected objective functions were used to find which one fit the data the best. Since then, a number of similar studies have appeared [13,15,16,19], along with the systematic evaluation of the space of all objective functions that match experimental data [14,17,18].

The cumulative data suggests that strains, such as the widely studied *E. coli* strains, that have been grown over long periods of time in laboratory settings, have acquired an optimal growth phenotype on commonly used substrates in growth media [26]. When confronted with an unfamiliar substrate, optimal growth phenotypes can be generated using laboratory adaptive evolution [27-30]. Evolved strains can then be re-sequenced to find all mutations generated, thus illuminating the underlying genetic and molecular biological basis for optimal growth phenotypes [31,32].

Nutritionally rich environments are probably the exception rather than the norm in natural environments. Thus, the studies just described may represent exceptions rather than the norm. In general, we might begin to conceptualize cellular survival strategies in order to formulate useful objective functions. Consider three different environments; 1) nutritionally rich, as above, 2) scarce nutritional environment, and 3) elementally limited environment. From a natural habitat standpoint and the experiences of microorganisms, these are perhaps listed from the least likely to the likeliest; however, no computational studies of the third case have appeared. For the first and second cases, data from batch growth (nutritionally rich, case one) and chemostat growth experiments (nutritionally scarce, case two) suggests that optimal biomass yield or growth rates are meaningful objectives [11-14,16,17,19]. However, cases have appeared indicating contrary objectives, such as maximization of ATP per unit flux, being better predictors of experimental data [16]. Nonetheless, maximal growth rate phenotype can still result after adaptive evolution, or through prolonged experimentation in the laboratory. It should be noted that a predictable phenomena becomes the basis for design. For example, growth coupling of a bioengineering production objective has emerged as a strain design strategy [33-36], with adaptive evolution being a tool to produce such designs [37].

The constraint-based formalism has been shown to work at the genome-scale [2,3]. It obviates the need for many details by incorporating an objective function and assuming optimal organism functions. Although, ‘everything in biology should be viewed through the eyes of evolution’ implies some optimal performance based on the organism's past history, we are only beginning to decipher what cellular objectives actually are. One can therefore anticipate that many studies of the objective function are to appear.

## Acknowledgments

We would like to thank Jacob D. Feala and Daniel C. Zielinski for their valuable feedback on this manuscript.

## Footnotes

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