特集 SFC バイオの軌跡［招待論文：総説・レビュー論文］

The Historical Evolution of
Mathematical Modeling of Metabolism
in the Human Red Blood Cell
ヒト赤血球代謝数理モデリングの歴史的発展
Zachary B. Haiman

Graduate Research Assistant at Department of Bioengineering, University of California San
Diego, La Jolla, CA

ザッカリー B ハイマン
カリフォルニア大学サンディエゴ校生物工学部大学院リサーチアシスタント

James T. Yurkovich

Visiting Scholar at Department of Bioengineering, University of California San Diego, La Jolla,CA;
Chief Innovation Officer at Phenome Health, Seattle, WAジェームス T ユルコビッチ
カリフォルニア大学サンディエゴ校生物工学部訪問研究員
Phenome Health チーフ・イノベーション・オフィサー

Bernhard O. Palsson

Y.C. Fung Endowed Chair and Professor at Department of Bioengineering, University ofCalifornia San Diego, La Jolla, CA

ベルナード O パルソン
カリフォルニア大学サンディエゴ校生物工学部 Y.C. Fung 寄付講座教授
Correspondence to: palsson@ucsd.edu

Abstract:

During the last half century significant advances in metabolic modeling andwhole-cell simulation using the human red blood cell as a model system wereachieved. From initial studies chaining enzymes together as pathways, modelingframeworks for metabolic pathways were developed. Simulation software capableof modeling multiple biochemical and genetic processes facilitated the move towardcell-scale modeling and enabled complementary workflows linking omic data andcomputational studies for model-driven design. Here, we review selected milestonesin the history of mathematical modeling of the human red blood cell.

8【第 1 章 先端生命科学】
The Historical Evolution of Mathematical Modeling of Metabolism in the Human Red Blood Cell
ヒト赤血球をモデル系とした代謝モデリングと全細胞シミュレーションはこの半世紀の間に大きく発展した。酵素をパスウェイとして連結する初期の研究に基づき、代謝パスウェイのモデリングフレームワークが開発されるようになった。複数の生化学的・遺伝学的プロセスをモデリングできるシミュレーショ
ンソフトウェアは、細胞スケールのモデリングへの移行を促進し、モデル駆動
設計のためのオミックスデータと計算機研究を結びつける補完的なワークフ
ローを可能にした。本稿では、ヒト赤血球の数理モデリングの歴史におけるマ
イルストーンを紹介する。

Keywords: red blood cell, metabolism, computer simulation, mathematical models, systems
biology

赤血球、代謝、コンピュータシミュレーション、数理モデル、システム生物学

Throughout the history of computational biochemical modeling, the human red
blood cell (RBC) has been an important model system. As the most abundant cell in
the human body (Sender et al., 2016), the RBC plays important physiological roles ingas transport (Doctor and Stamler, 2011; Garby and Meldon, 1977). Historically, the
RBC has been considered to be a relatively simple cell type whose main function wasin transport of respiratory gasses from the lung to the tissues. With the advent of omic
technologies, the RBC can now be viewed in a new light, showing that its
functionality may be more complex and essential than previously believed (Nemkov
et al., 2018). For almost five decades, the human RBC has served as the ideal cell
type for the development of mathematical models due to the significant amount ofavailable experimental data, ease of sampling, its relative metabolic simplicity, andits physiological robustness over the course of its 120-day lifespan. It has thus played
an important role in the history of systems biology and the quest to construct wholecell dynamic simulators.
Here, we review the history of mathematical modeling in the RBC. This history
breaks down into roughly four time periods (Fig. 1). We discuss a few selected
milestones from each period. We start by surveying early efforts to model enzyme
kinetics for individual enzymes and how they are chained together into pathway
models. As the scope and complexity of metabolic pathway models grew, theybrought about the period where whole-cell biophysical models were formulated. The
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Figure 1: A timeline of selected milestones in the history of modeling human red blood
cells. The history of whole-cell modeling for the human red blood cell is decades long, withmany researchers contributing pioneering work in the fields of biomedical science andcomputational systems biology. For the purposes of this review, we have broken the timeline
into four epochs which will structure our discussion: (1) Enzyme kinetics to metabolic
pathways; (2) Metabolic pathways to whole-cell biophysical models; (3) Whole-cell biophysicalmodels to omics-derived models; and (4) The next frontiers.

advent of high-throughput data collection methods made it possible to derive largescale models from omic data, signifying the start of the current era in which wholecell models are combined with omics data. In each epoch, RBC modeling has been
integral to several significant advancements in systems biology and biomedical
research. Our discussion aims to contextualize these important milestones and explore
how they may bridge the gap for future progress in whole-cell modeling of human
RBCs.

1

From Enzyme Kinetics to Metabolic Pathways
Early in the 20th Century, enzyme kinetics were developed and studied via well-controlled in vitro environments. These early efforts led to the formulation of classicalmechanisms for representing enzyme kinetics (Briggs and Haldane, 1925; Michaelis10

【第 1 章 先端生命科学】
The Historical Evolution of Mathematical Modeling of Metabolism in the Human Red Blood Cellet al., 2011). The formulation of new methods for deriving rate laws greatly reduced
the amount of time and effort required to mathematically model complicated enzyme
mechanisms (Cleland, 1967; King and Altman, 1956). Because the RBC is primarily
dependent on glycolysis for energy, it became the cell type of choice for many earlystudies of enzyme and pathway kinetics (Heinrich et al., 1978; Heinrich andRapoport, 1974a). Furthermore, the study of allosteric binding—like oxygen binding
to hemoglobin—led to the development of models representing the kinetics
associated with allosteric effectors (Monod et al., 1965).
The recognition that computers could represent an opportunity to study multienzyme systems spurred the development of programs for the digital representationof chemical reactions (Chance et al., 1960). Starting with simple chemical kinetics,
computational methods quickly evolved to simulate complex enzyme mechanisms(Garfinkel et al., 1961, 1966). Consequently, these developments led to single-enzyme
systems that were computationally modeled with significant detail, enabling an
exploration of the feasibility for proposed enzymatic mechanisms as well as the
elucidation of new mechanistic insights through the fitting of results to match
experimental data.
Single-enzyme kinetics could be modeled with increased detail and accuracy
due to the smaller number of parameters and the need to only fit one reaction velocity
to experimental data. However, modeling multi-enzyme systems proved to besignificantly more complicated. Additional complexities arise when considering
enzymes that are influenced by effectors simultaneously, and enzymes that are
affected by compartmentalization and cellular membranes (Heinrich et al., 1978). To
analyze features of regulation of a metabolic pathway, Heinrich and Rapoport(1974b) used control theory to generate a linear approximation of multi-enzyme
systems. They utilized a steady-state assumption and simplified differential equations
to consider the several enzymes which, when chained together, form a pathway (Fig.
2). By considering the entire pathway, identification of effector binding sites in realsystems could be obtained while avoiding the erroneous conclusions that were drawn
through the simple crossover theorem (Heinrich and Rapoport, 1974a).
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Figure 2: Application of the steady state assumption to a pathway of enzymes. (Rapoport etal., 1974) developed a framework for mathematical analyses of regulatory control in metabolic
pathways by assuming the influxes and effluxes for a given metabolite are balanced at steadystate. The application of the control theory of steady states to the glycolytic pathway inerythrocytes revealed that f lux is primarily controlled by the hexokinase andphosphofructokinase enzymes (Rapoport et al., 1974). Abbreviations: GLC, glucose; G6P,
glucose 6-phosphate; F6P, fructose 6-phosphate; FDP, fructose 1,6-bisphosphate; HK,
hexokinase; PGI, phosphoglucoisomerase; PFK, phosphofructokinase.

By applying steady-state kinetics, Rapoport et al. (1974) derived a pathway
model of glycolysis in human red blood cells. Mathematical analyses of thecharacteristic time constants revealed that glycolytic flux is controlled by thehexokinase (HK) and phosphofructokinase (PFK) enzymes, and that control was
indicative of enzymatic interactions with effector molecules (Fig.2). Subsequent
expansion of the model included the adenosine triphosphate (ATP) synthesis anddegradation, demonstrating a buffering effect produced by the Rapoport-Luebering
shunt for maintaining the ATP concentration (Rapoport and Heinrich, 1975). Schauer
et al. (1981) endeavored to explore the functional roles of adenosine nucleotide
metabolism by including synthesis and degradation of adenine nucleotides in their
expanded model. Ataullakhanov et al. (1981) connected the glycolytic and pentose
phosphate pathways and analyzed the interdependence of pathway activities withrespect to stabilization of key cofactor concentrations. To quantify the interlinked12【第 1 章 先端生命科学】
The Historical Evolution of Mathematical Modeling of Metabolism in the Human Red Blood Cell

processes of energy metabolism and osmoregulation, Brumen and Heinrich (1984)
incorporated active and passive ion transport processes into the glycolytic pathway
model and formed a metabolic osmotic model.
Through the application of expanded glycolytic models, mathematical modelingwas shown to be capable of quantitatively estimating metabolic consequences of anindividual enzyme afflicted with an enzymopathy. Using calculated kinetic
parameters obtained from three patients afflicted with varying severities of pyruvatekinase (PYK) deficiency, Holzhutter et al. (1985) validated previously held
assumptions concerning the severity of hemolytic anemia and metabolic
abnormalities in energy metabolism. This study exemplified how mathematicalmodels could represent metabolic abnormalities consistent with clinical phenotypes.
With advances in computing hardware and theories for biological modeling (Lumb,
1987)—combined with improvements in DNA sequencing automation (Smith et al.,1986)—increasing the scope, scale, and complexity of biochemical models was
becoming ever more possible.

2

From Metabolic Pathways to Whole-cell Biophysical Models
The various expansions of the glycolytic pathway model enabled thequantification and prediction of physiological phenomena through separate
mathematical models. However, an integrated model of all these pathways waslacking. In 1989, Joshi and Palsson built a comprehensive cell-scale kinetic model ofthe RBC in which glycolysis, the pentose phosphate pathway, adenine nucleotide
metabolism, osmotic pressure balancing, electroneutrality, and transport processes
were all included in one unified modeling framework. They proceeded to demonstrate
the practical utility of such a whole-cell model through various simulations thatpredicted the pH dependence of the Donnan ratio, the consequences of PYK genetic
variation, and the dynamic responses of key compounds under blood storage
conditions (Joshi and Palsson, 1990).
The increasing scope and complexity of metabolic models—in which several
cellular processes interact in various manners across multiple time-scales—
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特集 SFC バイオの軌跡

necessitated the development of multipurpose simulation software capable of wholecell simulations. Extensible software that adheres to an object-oriented modelingparadigm was needed to ensure accuracy when modeling at the scale of whole-cell
RBC metabolism (Takahashi et al., 2002; Yachie-Kinoshita et al., 2010). To addressthese challenges, Tomita et al. (1999) created the E-CELL software environment, thefirst framework for whole cell modeling of cellular processes. Using the E-CELL
framework, interactive and repeatable simulations can be conducted to facilitate awide variety of in-silico experiments. The visualization and analysis of generated
predictions could then serve as complementary guides for subsequent experimental
design and interpretation in model-driven design (Fig. 3).
Through successive improvements, E-CELL became a multi-platform
framework for whole-cell simulation (Takahashi et al., 2003). As one of the first to
include the Systems Biology Mark-up Language (SBML), E-CELL helped initiate
the community-wide adoption of standards for model interoperability andexchangeability (Hucka et al., 2003). Following the emergence of parallel computing,the E-CELL simulation engine was redesigned for scalable modeling. The
implementation of a modular meta-algorithm for mixed-mode simulation enabled
various time-driven algorithms to be assigned to different model components,
facilitating the simulation of cellular subsystems interacting across multiple
timescales (Takahashi et al., 2002; 2004).
The ability to reproduce cellular behavior through repeatable simulations forvarious conditions was foreseen as having significant benefits in understanding theeffects of abnormal pathological conditions (Tomita, 2001). For example, single
nucleotide polymorphisms (SNPs) could result in different enzymes and thevariations in their kinetic parameters. Simulations demonstrated how the genotype
associated with an enzymopathy and the manifestation of its clinical phenotype could
be understood in the context of cellular metabolism (Jamshidi et al., 2002). The study
characterized the cellular responses for several variants of both PYK and G6PDH
deficiencies to elevated metabolic loads and elucidated insights into differences incellular function for chronic and non chronic hemolytic anemia.

14

【第 1 章 先端生命科学】
The Historical Evolution of Mathematical Modeling of Metabolism in the Human Red Blood Cell

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Figure 3: The cyclic process of model-driven design and discovery. Biochemical knowledge
and experimental findings are integrated to formulate hypotheses that can be explored withmathematical models. Using multipurpose, multiplatform software environments for scalable
simulation, mathematical models are constructed and simulated to run in silico experiments topredict cellular responses for different conditions (boxed with solid lines). Predictions are thenvisualized and analyzed to understand the systematic responses of cellular components andgenerate hypotheses. Using the knowledge derived from experimental validation of hypotheses,models are refined and expanded for subsequent in silico studies in the cyclic process of modeldriven design. The metabolic map shown is based on the refinement and expansion of the cellscale RBC model (Joshi and Palsson, 1989) to include de novo glutathione synthesis and export(Nakayama et al., 2005), magnesium binding, hemoglobin allostery, Band 3 interactions(Kinoshita et al., 2007; Nishino et al., 2009), and guanosine uptake (Nishino et al., 2013).
Abbreviations: GLC, glucose; GSSG, oxidized glutathione, HX, hypoxanthine; ADE, adenine,
Pi, inorganic phosphate; ADO, adenosine; INO, inosine; LAC, lactate; PYR, pyruvate; Na,sodium ion; K, potassium ion; Mg, magnesium ion; PFK, phosphofructokinase; ALD, aldolase;
GAPDH, glyceraldehyde-3-phosphate dehydrogenase;, oxyHb, oxyhemoglobin; deoxyHb,
deoxyhemoglobin; B3, Band 3 membrane protein.
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Discrepancies between in silico predictions and experimentally determined
observations are inevitable, making algorithmic assessment and iterative refinementof models a necessary and critical step for their improvement (Ni and Savageau,1996). When simulating a variant of glucose-6-phosphate dehydrogenase (G6PDH)
deficiency, Nakayama et al. (2005) found that their initial model predicted asignificantly shorter half-life compared to real cells containing the same deficiency.
By refining their model via the inclusion of pathways for de novo synthesis of
reduced glutathione (GSH) and export of oxidized glutathione (GSSG), theysuccessfully demonstrated the essentiality of these pathways for accurate simulationof G6PDH deficiency. Furthermore, they showed that abnormal conditions such as
those caused by enzymopathies may result in increased activation of typically lowactivity subsystems as a compensatory mechanism. The broad implications of thesefindings are that the inclusion of many, if not all, metabolic pathways is necessary forthe accurate simulation of metabolic abnormalities. Ultimately, the research during
this time resulted in scalable models and computing frameworks that allowed for thesimulation of large-scale models.

3

From Whole-cell Biophysical Models to Omics-derived
Models
The availability of scalable simulation environments lowered the technologicalbarriers associated with large-scale dynamic modeling and simulation. Genome
sequences could be utilized in automating construction of large metabolic models,
reducing the manual labor required to formulate genome-scale metabolic models(Arakawa et al., 2006). Thus, limitations in dynamic modeling evolved to become
primarily associated with parameterization due to the considerable amount of datarequired for such large-scale models. Mathematical frameworks for analyzing
metabolic flux states through constraint-based reconstruction and analysis (COBRA)
were shown to be effective at large-scale analysis of metabolic networks (Price et al.,2004). COBRA models rely primarily on network structure derived from genomics
and flux balance analysis (FBA), requiring minimal parameterization for accurate16

【第 1 章 先端生命科学】
The Historical Evolution of Mathematical Modeling of Metabolism in the Human Red Blood Cell

predictions (Varma and Palsson, 1994). As a model cell type, the human RBC model
was also utilized in development of many of these novel approaches for addressing
the sparsity of kinetic data using omic data to reduce the number of kinetic
parameters required for dynamic simulation (Yugi et al., 2005). Using genomic andproteomic data, Bordbar et al. (2011) derived one of the most expansive cell-scalereconstructions of RBC metabolism. By mapping morbid SNPs and pharmaceuticaltreatments that are known to target the RBC and by applying flux variability analysis,
the RBC was shown to have a more extensive role in human metabolism thanpreviously believed, confirming its potential as a clinical biomarker.
Other omic data like metabolomics have become a useful tool for studying themetabolic state of cellular systems. The technological innovations surrounding highthroughput methods made it possible to collect metabolomics data for both qualitativeand qualitative analyses (Soga et al., 2003, 2004). Accordingly, it became possible toverify predictions made with metabolic models through analysis of the metabolome.
Circulating RBCs are constantly subject to oxidative stress, undergoing changes
between normoxic and hypoxic conditions, and evidence suggested Band 3
membrane protein interactions with hemoglobin may trigger compensatory
mechanisms for maintaining intracellular ATP levels in hypoxic RBCs. Researchersbegan to study this phenomenon using a dynamic model containing Band 3interactions with hemoglobin and glycolytic enzymes (Kinoshita et al., 2007). The
comparison of in silico predictions with the results obtained from metabolome
analyses provided mechanistic insight into how hemoglobin allostery and T-statebinding interactions with metabolites facilitated the maintenance of ATP and
2,3-Diphosphoglycerate (2,3-DPG) concentrations.
The findings by Kinoshita et al. (2007) had implications for RBCs in blood
storage, as there were no effective methods to prevent their ATP and 2,3-DPG
depletion. Using a genetic algorithm for parameter estimation (Kikuchi et al., 2003),
Nishino et al. (2009) modified the previous model to reflect RBCs preserved in coldstorage conditions. Through in silico experiments and metabolome analyses, they not
only verified their model but also identified several factors for improving blood
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storage conditions through maintenance of ATP and 2,3-DPG levels. By refining themetabolic model to include guanine uptake and the combination of in silico andmetabolome analyses, Nishino et al. (2013) subsequently provided clarity into themechanistic basis driving ATP and 2,3-DPG maintenance and revealed insights into
the trade-offs between metabolic benefits and side-effects associated with the additive
solution.
Other researchers also applied these modeling methods to blood storage as arelevant and actionable application of mathematical modeling tools. By integrating
the latest RBC reconstruction with time-course metabolomics data obtained from
RBCs in storage, metabolic states at various stages of RBC decay under storage
conditions were identified, leading to the discovery of key biomarkers that defined
the metabolic age of stored RBCs (Bordbar et al., 2016; Paglia et al., 2016). To
quantitatively predict dynamic intracellular metabolic changes of the network, anovel FBA method capable of reconciling time-course metabolomics and network
structure was developed. Subsequent application of the method provided new insights
into extracellular citrate as a storage additive that were not observable with standard
FBA methods (Bordbar et al., 2017; Yurkovich et al., 2017). The RBC metabolic
reconstruction (Bordbar et al., 2011) also served as the basis for the creation of
personalized RBC models. To study the metabolic consequences of individualized
variation, genomic and fasting-state metabolomic data was obtained and mapped onto
the constraint-based reconstruction (Bordbar et al., 2015).

4

The Next FrontiersSignificant advances in metabolic modeling and whole-cell simulation wereachieved over the past fifty years. Modeling of the RBC has been critical at each stepof the way: from the initial development of a framework for modeling metabolic
pathways in the 1970s to the recent applications in personalized modeling, biomarker
identification, and novel modeling method development. As whole-cell modeling is
critical to understanding the genotype-phenotype relationship, the increasing
recognition of its potential in transforming medicine is leading to the development of18

【第 1 章 先端生命科学】
The Historical Evolution of Mathematical Modeling of Metabolism in the Human Red Blood Cell

guiding principles in generating whole human cell dynamical models (Szigeti et al.,2018). New insights into the conservation of kinetic parameters across genome-scale
kinetic models offer the opportunity to understand the mechanisms of genotypephenotype relationships (Palsson and Yurkovich, 2022). With the level of detailprovided by longitudinal studies, detailed data-driven models of disease development
can be developed to enable precision medicine and personalized approaches.
The human RBC has been used as a model cell for the development of modelingmethods and driven the advancement of simulation technologies. These advances
have aided in the understanding of human metabolism and genetic variation on a
personalized level as these modeling approaches not only led to discoveries in the
RBC but were applied to other cellular systems. Notably, the broader development ofwhole-cell computable models—and all the accompanying computational methods,
simulation frameworks, and standards—has been fundamentally advanced by someof the work reviewed here that pioneered these tools in the RBC. The RBC willundoubtedly continue to play a significant role in developing complete proteomeconstrained models, personalized medicine, and other notable milestones yet to comein future work.
Several foundational developments drove the history of mathematical modelingof the human RBC. The development of the E-CELL platform—a pioneering
simulation platform that enables easy and informative simulation of whole cell
behavior—was an important milestone. As one of the first multi-platform softwareenvironments for standardized modeling, E-CELL provided the framework necessary
to facilitate the move toward cell-scale modeling by linking biochemical and genetic
processes to simulate cellular dynamics across multiple timescales. Further, its
existence catalyzed the development of other such modeling and simulation
platforms. Through these accomplishments, Tomita and colleagues exemplified how
metabolome analysis and in silico simulation could be paired together to understandunderlying metabolic mechanisms through whole-cell modeling and simulation.

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Source of support
This research was supported by funds from the Y.C. Fung Endowed Chair in
Bioengineering.

Conflict of interest
The authors declare no competing financial interests.

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