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Joshi B, Craciun G. Foundations of static and dynamic absolute concentration robustness. J Math Biol 2022; 85:53. [PMID: 36243796 DOI: 10.1007/s00285-022-01823-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2021] [Revised: 09/14/2022] [Accepted: 10/03/2022] [Indexed: 10/17/2022]
Abstract
Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg (Science 327:1389-1391, 2010) as robustness of equilibrium species concentration in a mass action dynamical system. Their aim was to devise a mathematical condition that will ensure robustness in the function of the biological system being modeled. The robustness of function rests on what we refer to as empirical robustness-the concentration of a species remains unvarying, when measured in the long run, across arbitrary initial conditions. Even simple examples show that the ACR notion introduced in Shinar and Feinberg (Science 327:1389-1391, 2010) (here referred to as static ACR) is neither necessary nor sufficient for empirical robustness. To make a stronger connection with empirical robustness, we define dynamic ACR, a property related to long-term, global dynamics, rather than only to equilibrium behavior. We discuss general dynamical systems with dynamic ACR properties as well as parametrized families of dynamical systems related to reaction networks. We find necessary and sufficient conditions for dynamic ACR in complex balanced reaction networks, a class of networks that is central to the theory of reaction networks.
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Affiliation(s)
- Badal Joshi
- Department of Mathematics, California State University San Marcos, San Marcos, USA.
| | - Gheorghe Craciun
- Departments of Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison, Madison, USA
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2
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Lubenia PVN, Mendoza ER, Lao AR. Reaction Network Analysis of Metabolic Insulin Signaling. Bull Math Biol 2022; 84:129. [PMID: 36168001 DOI: 10.1007/s11538-022-01087-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2022] [Accepted: 09/15/2022] [Indexed: 11/02/2022]
Abstract
Absolute concentration robustness (ACR) and concordance are novel concepts in the theory of robustness and stability within Chemical Reaction Network Theory. In this paper, we have extended Shinar and Feinberg's reaction network analysis approach to the insulin signaling system based on recent advances in decomposing reaction networks. We have shown that the network with 20 species, 35 complexes, and 35 reactions is concordant, implying at most one positive equilibrium in each of its stoichiometric compatibility class. We have obtained the system's finest independent decomposition consisting of 10 subnetworks, a coarsening of which reveals three subnetworks which are not only functionally but also structurally important. Utilizing the network's deficiency-oriented coarsening, we have developed a method to determine positive equilibria for the entire network. Our analysis has also shown that the system has ACR in 8 species all coming from a deficiency zero subnetwork. Interestingly, we have shown that, for a set of rate constants, the insulin-regulated glucose transporter GLUT4 (important in glucose energy metabolism), has stable ACR.
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Affiliation(s)
- Patrick Vincent N Lubenia
- Systems and Computational Biology Research Unit, Center for Natural Sciences and Environmental Research, 2401 Taft Avenue, Manila, 0922, Metro Manila, Philippines
| | - Eduardo R Mendoza
- Systems and Computational Biology Research Unit, Center for Natural Sciences and Environmental Research, 2401 Taft Avenue, Manila, 0922, Metro Manila, Philippines.,Department of Mathematics and Statistics, De La Salle University, 2401 Taft Avenue, Manila, 0922, Metro Manila, Philippines.,Max Planck Institute of Biochemistry, Am Klopferspitz 18, 82152, Martinsried, Munich, Germany
| | - Angelyn R Lao
- Systems and Computational Biology Research Unit, Center for Natural Sciences and Environmental Research, 2401 Taft Avenue, Manila, 0922, Metro Manila, Philippines. .,Department of Mathematics and Statistics, De La Salle University, 2401 Taft Avenue, Manila, 0922, Metro Manila, Philippines. .,Center for Complexity and Emerging Technologies, 2401 Taft Avenue, Manila, 0922, Metro Manila, Philippines.
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Nam KM, Martinez-Corral R, Gunawardena J. The linear framework: using graph theory to reveal the algebra and thermodynamics of biomolecular systems. Interface Focus 2022; 12:20220013. [PMID: 35860006 PMCID: PMC9184966 DOI: 10.1098/rsfs.2022.0013] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2022] [Accepted: 04/25/2022] [Indexed: 12/25/2022] Open
Abstract
The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent biochemical species or molecular states, edges represent reactions or transitions and labels represent rates. The graph yields a linear dynamics for molecular concentrations or state probabilities, with the graph Laplacian as the operator, and the labels encode the nonlinear interactions between system and environment. The labels can be specified by vertices of other graphs or by conservation laws or, when the environment consists of thermodynamic reservoirs, they may be constants. In the latter case, the graphs correspond to infinitesimal generators of Markov processes. The key advantage of the framework has been that steady states are determined as rational algebraic functions of the labels by the Matrix-Tree theorems of graph theory. When the system is at thermodynamic equilibrium, this prescription recovers equilibrium statistical mechanics but it continues to hold for non-equilibrium steady states. The framework goes beyond other graph-based approaches in treating the graph as a mathematical object, for which general theorems can be formulated that accommodate biomolecular complexity. It has been particularly effective at analysing enzyme-catalysed modification systems and input-output responses.
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Affiliation(s)
- Kee-Myoung Nam
- Department of Systems Biology, Harvard Medical School, Boston, MA 02115, USA
| | | | - Jeremy Gunawardena
- Department of Systems Biology, Harvard Medical School, Boston, MA 02115, USA
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Cappelletti D, Gupta A, Khammash M. A hidden integral structure endows absolute concentration robust systems with resilience to dynamical concentration disturbances. J R Soc Interface 2020; 17:20200437. [PMID: 33109021 DOI: 10.1098/rsif.2020.0437] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023] Open
Abstract
Biochemical systems that express certain chemical species of interest at the same level at any positive steady state are called 'absolute concentration robust' (ACR). These species behave in a stable, predictable way, in the sense that their expression is robust with respect to sudden changes in the species concentration, provided that the system reaches a (potentially new) positive steady state. Such a property has been proven to be of importance in certain gene regulatory networks and signaling systems. In the present paper, we mathematically prove that a well-known class of ACR systems studied by Shinar and Feinberg in 2010 hides an internal integral structure. This structure confers these systems with a higher degree of robustness than was previously known. In particular, disturbances much more general than sudden changes in the species concentrations can be rejected, and robust perfect adaptation is achieved. Significantly, we show that these properties are maintained when the system is interconnected with other chemical reaction networks. This key feature enables the design of insulator devices that are able to buffer the loading effect from downstream systems-a crucial requirement for modular circuit design in synthetic biology. We further note that while the best performance of the insulators are achieved when these act at a faster timescale than the upstream module (as typically required), it is not necessary for them to act on a faster timescale than the downstream module in our construction.
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Affiliation(s)
- Daniele Cappelletti
- Department of Biosystems Science and Engineering, ETH Zurich, Mattenstrasse 26 4058 Basel, Switzerland
| | - Ankit Gupta
- Department of Biosystems Science and Engineering, ETH Zurich, Mattenstrasse 26 4058 Basel, Switzerland
| | - Mustafa Khammash
- Department of Biosystems Science and Engineering, ETH Zurich, Mattenstrasse 26 4058 Basel, Switzerland
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Krishnan J, Lu L, Alam Nazki A. The interplay of spatial organization and biochemistry in building blocks of cellular signalling pathways. J R Soc Interface 2020; 17:20200251. [PMID: 32453980 PMCID: PMC7276544 DOI: 10.1098/rsif.2020.0251] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2020] [Accepted: 04/24/2020] [Indexed: 12/14/2022] Open
Abstract
Biochemical pathways and networks are central to cellular information processing. While a broad range of studies have dissected multiple aspects of information processing in biochemical pathways, the effect of spatial organization remains much less understood. It is clear that space is central to intracellular organization, plays important roles in cellular information processing and has been exploited in evolution; additionally, it is being increasingly exploited in synthetic biology through the development of artificial compartments, in a variety of ways. In this paper, we dissect different aspects of the interplay between spatial organization and biochemical pathways, by focusing on basic building blocks of these pathways: covalent modification cycles and two-component systems, with enzymes which may be monofunctional or bifunctional. Our analysis of spatial organization is performed by examining a range of 'spatial designs': patterns of localization or non-localization of enzymes/substrates, theoretically and computationally. Using these well-characterized in silico systems, we analyse the following. (i) The effect of different types of spatial organization on the overall kinetics of modification, and the role of distinct modification mechanisms therein. (ii) How different information processing characteristics seen experimentally and studied from the viewpoint of kinetics are perturbed, or generated. (iii) How the activity of enzymes (bifunctional enzymes in particular) may be spatially manipulated, and the relationship between localization and activity. (iv) How transitions in spatial organization (encountered either through evolution or through the lifetime of cells, as seen in multiple model organisms) impacts the kinetic module (and pathway) behaviour, and how transitions in chemistry may be impacted by prior spatial organization. The basic insights which emerge are central to understanding the role of spatial organization in biochemical pathways in both bacteria and eukaryotes, and are of direct relevance to engineering spatial organization of pathways in bottom-up synthetic biology in cellular and cell-free systems.
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Affiliation(s)
- J. Krishnan
- Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
- Institute for Systems and Synthetic Biology, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
| | - Lingjun Lu
- Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
| | - Aiman Alam Nazki
- Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
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Monostationarity and Multistationarity in Tree Networks of Goldbeter-Koshland Loops. Bull Math Biol 2019; 81:2463-2509. [PMID: 31218553 DOI: 10.1007/s11538-019-00615-y] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2017] [Accepted: 05/21/2019] [Indexed: 10/26/2022]
Abstract
A major challenge in systems biology is to elicit general properties in the face of molecular complexity. Here, we introduce a class of enzyme-catalysed biochemical networks and examine how the existence of a single positive steady state (monostationarity) depends on the network structure, enzyme mechanisms, kinetic rate laws and parameter values. We consider Goldbeter-Koshland (GK) covalent modification loops arranged in a tree network, so that a substrate form in one loop can be an enzyme in another loop. GK loops are a canonical motif in cell signalling and trees offer a generalisation of linear cascades which accommodate network complexity while remaining mathematically tractable. In particular, they permit a modular, recursive proof strategy which may be more widely applicable. We show that if each enzyme follows its own complex reaction mechanism under mass action kinetics, then any network is monostationary for all appropriate parameter values. If the kinetics is non-mass action with a plausible monotonicity requirement, and each enzyme follows the Michaelis-Menten mechanism, then monostationarity is preserved. Surprisingly, a single GK loop with a complex enzyme mechanism under non-mass action monotone kinetics can have more than one positive steady state (multistationarity). The broader interplay between network structure, enzyme mechanism and kinetics remains an intriguing open problem.
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Straube R. Analysis of network motifs in cellular regulation: Structural similarities, input-output relations and signal integration. Biosystems 2017; 162:215-232. [PMID: 29107640 DOI: 10.1016/j.biosystems.2017.10.012] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2017] [Revised: 10/22/2017] [Accepted: 10/23/2017] [Indexed: 02/07/2023]
Abstract
Much of the complexity of regulatory networks derives from the necessity to integrate multiple signals and to avoid malfunction due to cross-talk or harmful perturbations. Hence, one may expect that the input-output behavior of larger networks is not necessarily more complex than that of smaller network motifs which suggests that both can, under certain conditions, be described by similar equations. In this review, we illustrate this approach by discussing the similarities that exist in the steady state descriptions of a simple bimolecular reaction, covalent modification cycles and bacterial two-component systems. Interestingly, in all three systems fundamental input-output characteristics such as thresholds, ultrasensitivity or concentration robustness are described by structurally similar equations. Depending on the system the meaning of the parameters can differ ranging from protein concentrations and affinity constants to complex parameter combinations which allows for a quantitative understanding of signal integration in these systems. We argue that this approach may also be extended to larger regulatory networks.
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Affiliation(s)
- Ronny Straube
- Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Sandtorstr. 1, D-39106 Magdeburg, Germany
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Straube R. Operating regimes of covalent modification cycles at high enzyme concentrations. J Theor Biol 2017; 431:39-48. [PMID: 28782551 DOI: 10.1016/j.jtbi.2017.08.006] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2017] [Revised: 08/02/2017] [Accepted: 08/03/2017] [Indexed: 10/19/2022]
Abstract
The Goldbeter-Koshland model has been a paradigm for ultrasensitivity in biological networks for more than 30 years. Despite its simplicity the validity of this model is restricted to conditions when the substrate is in excess over the converter enzymes - a condition that is easy to satisfy in vitro, but which is rarely satisfied in vivo. Here, we analyze the Goldbeter-Koshland model by means of the total quasi-steady state approximation which yields a comprehensive classification of the steady state operating regimes under conditions when the enzyme concentrations are comparable to or larger than that of the substrate. Where possible we derive simple expressions characterizing the input-output behavior of the system. Our analysis suggests that enhanced sensitivity occurs if the concentration of at least one of the converter enzymes is smaller (but not necessarily much smaller) than that of the substrate and if that enzyme is saturated. Conversely, if both enzymes are saturated and at least one of the enzyme concentrations exceeds that of the substrate the system exhibits concentration robustness with respect to changes in that enzyme concentration. Also, depending on the enzyme's saturation degrees and the ratio between their maximal reaction rates the total fraction of phosphorylated substrate may increase, decrease or change nonmonotonically as a function of the total substrate concentration. The latter finding may aid the interpretation of experiments involving genetic perturbations of enzyme and substrate abundances.
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Affiliation(s)
- Ronny Straube
- Department of Mathematics, Friedrich-Alexander-University Erlangen-Nuernberg, Cauerstr. 11, D-91058 Erlangen, Germany.
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Kuwahara H, Umarov R, Almasri I, Gao X. ACRE: Absolute concentration robustness exploration in module-based combinatorial networks. Synth Biol (Oxf) 2017; 2:ysx001. [PMID: 32995502 PMCID: PMC7513739 DOI: 10.1093/synbio/ysx001] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2016] [Revised: 12/03/2016] [Accepted: 01/03/2017] [Indexed: 11/26/2022] Open
Abstract
To engineer cells for industrial-scale application, a deep understanding of how to design molecular control mechanisms to tightly maintain functional stability under various fluctuations is crucial. Absolute concentration robustness (ACR) is a category of robustness in reaction network models in which the steady-state concentration of a molecular species is guaranteed to be invariant even with perturbations in the other molecular species in the network. Here, we introduce a software tool, absolute concentration robustness explorer (ACRE), which efficiently explores combinatorial biochemical networks for the ACR property. ACRE has a user-friendly interface, and it can facilitate efficient analysis of key structural features that guarantee the presence and the absence of the ACR property from combinatorial networks. Such analysis is expected to be useful in synthetic biology as it can increase our understanding of how to design molecular mechanisms to tightly control the concentration of molecular species. ACRE is freely available at https://github.com/ramzan1990/ACRE.
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Affiliation(s)
- Hiroyuki Kuwahara
- Computational Bioscience Research Center (CBRC), King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia.,Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia
| | - Ramzan Umarov
- Computational Bioscience Research Center (CBRC), King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia.,Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia
| | - Islam Almasri
- Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia
| | - Xin Gao
- Computational Bioscience Research Center (CBRC), King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia.,Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia
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Gu C, Nguyen HN, Hofer A, Jessen HJ, Dai X, Wang H, Shears SB. The Significance of the Bifunctional Kinase/Phosphatase Activities of Diphosphoinositol Pentakisphosphate Kinases (PPIP5Ks) for Coupling Inositol Pyrophosphate Cell Signaling to Cellular Phosphate Homeostasis. J Biol Chem 2017; 292:4544-4555. [PMID: 28126903 PMCID: PMC5377771 DOI: 10.1074/jbc.m116.765743] [Citation(s) in RCA: 51] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2016] [Revised: 01/25/2017] [Indexed: 12/31/2022] Open
Abstract
Proteins responsible for Pi homeostasis are critical for all life. In Saccharomyces cerevisiae, extracellular [Pi] is "sensed" by the inositol-hexakisphosphate kinase (IP6K) that synthesizes the intracellular inositol pyrophosphate 5-diphosphoinositol 1,2,3,4,6-pentakisphosphate (5-InsP7) as follows: during a period of Pi starvation, there is a decline in cellular [ATP]; the unusually low affinity of IP6Ks for ATP compels 5-InsP7 levels to fall in parallel (Azevedo, C., and Saiardi, A. (2017) Trends. Biochem. Sci. 42, 219-231. Hitherto, such Pi sensing has not been documented in metazoans. Here, using a human intestinal epithelial cell line (HCT116), we show that levels of both 5-InsP7 and ATP decrease upon [Pi] starvation and subsequently recover during Pi replenishment. However, a separate inositol pyrophosphate, 1,5-bisdiphosphoinositol 2,3,4,6-tetrakisphosphate (InsP8), reacts more dramatically (i.e. with a wider dynamic range and greater sensitivity). To understand this novel InsP8 response, we characterized kinetic properties of the bifunctional 5-InsP7 kinase/InsP8 phosphatase activities of full-length diphosphoinositol pentakisphosphate kinases (PPIP5Ks). These data fulfil previously published criteria for any bifunctional kinase/phosphatase to exhibit concentration robustness, permitting levels of the kinase product (InsP8 in this case) to fluctuate independently of varying precursor (i.e. 5-InsP7) pool size. Moreover, we report that InsP8 phosphatase activities of PPIP5Ks are strongly inhibited by Pi (40-90% within the 0-1 mm range). For PPIP5K2, Pi sensing by InsP8 is amplified by a 2-fold activation of 5-InsP7 kinase activity by Pi within the 0-5 mm range. Overall, our data reveal mechanisms that can contribute to specificity in inositol pyrophosphate signaling, regulating InsP8 turnover independently of 5-InsP7, in response to fluctuations in extracellular supply of a key nutrient.
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Affiliation(s)
- Chunfang Gu
- From the Laboratory of Signal Transduction, NIEHS, National Institutes of Health, Research Triangle Park, North Carolina, 27709
| | - Hoai-Nghia Nguyen
- From the Laboratory of Signal Transduction, NIEHS, National Institutes of Health, Research Triangle Park, North Carolina, 27709
| | - Alexandre Hofer
- Department of Chemistry, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
| | - Henning J Jessen
- Institute of Organic Chemistry, Albert Ludwigs University, Albertstrasse 21, 79104 Freiburg, Germany, and
| | - Xuming Dai
- Division of Cardiology, McAllister Heart Institute, University of North Carolina, Chapel Hill, North Carolina 27599
| | - Huanchen Wang
- From the Laboratory of Signal Transduction, NIEHS, National Institutes of Health, Research Triangle Park, North Carolina, 27709
| | - Stephen B Shears
- From the Laboratory of Signal Transduction, NIEHS, National Institutes of Health, Research Triangle Park, North Carolina, 27709,
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