Catalog of MCA Control Patterns

7213: 5602: 7208:{\displaystyle {\begin{array}{llll}C_{v_{1}}^{J}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{v_{2}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{v_{3}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{v_{4}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{3}-\varepsilon _{3}^{1}\varepsilon _{1}^{2}\varepsilon _{2}^{3}\\C_{v_{1}}^{S_{1}}=\varepsilon _{2}^{2}\varepsilon _{3}^{3}-\varepsilon _{2}^{2}\varepsilon _{3}^{4}+\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{v_{2}}^{S_{1}}=\varepsilon _{3}^{1}\varepsilon _{2}^{3}-\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{v_{3}}^{S_{1}}=-\varepsilon _{3}^{1}\varepsilon _{2}^{2}+\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{v_{4}}^{S_{1}}=\varepsilon _{3}^{1}\varepsilon _{2}^{2}-\varepsilon _{3}^{1}\varepsilon _{2}^{3}-\varepsilon _{2}^{2}\varepsilon _{3}^{3}\\C_{v_{1}}^{S_{2}}=-\varepsilon _{1}^{2}\varepsilon _{3}^{3}+\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{v_{2}}^{S_{2}}=\varepsilon _{1}^{1}\varepsilon _{3}^{3}-\varepsilon _{1}^{1}\varepsilon _{3}^{4}&C_{v_{3}}^{S_{2}}=\varepsilon _{B}^{1}\varepsilon _{3}^{4}+\varepsilon _{3}^{1}\varepsilon _{1}^{2}-\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{v_{4}}^{S_{2}}=-\varepsilon _{1}^{1}\varepsilon _{3}^{3}-\varepsilon _{3}^{1}\varepsilon _{1}^{2}+\varepsilon _{1}^{2}\varepsilon _{3}^{3}\\C_{v_{1}}^{S_{3}}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{v_{2}}^{S_{3}}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{v_{3}}^{S_{3}}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}&C_{v_{4}}^{S_{3}}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}+\varepsilon _{1}^{1}\varepsilon _{2}^{3}-\varepsilon _{1}^{2}\varepsilon _{2}^{3}\end{array}}} 4494: 3123: 4489:{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=-\varepsilon _{1}^{2}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{2}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{3}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{e_{4}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{3}\\C_{e_{1}}^{s_{1}}=-\varepsilon _{2}^{2}\varepsilon _{3}^{3}+\varepsilon _{2}^{2}\varepsilon _{3}^{4}-\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}\varepsilon _{3}^{4}&C_{e_{3}}^{s_{1}}=-\varepsilon _{2}^{2}\varepsilon _{3}^{4}&C_{e_{4}}^{s_{1}}=\varepsilon _{2}^{2}\varepsilon _{3}^{3}\\C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}\varepsilon _{3}^{3}-\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{e_{2}}^{s_{2}}=-\varepsilon _{1}^{1}\varepsilon _{3}^{3}+\varepsilon _{1}^{1}\varepsilon _{3}^{4}&C_{e_{3}}^{s_{2}}=-\varepsilon _{1}^{1}\varepsilon _{3}^{4}+\varepsilon _{1}^{2}\varepsilon _{3}^{4}&C_{e_{4}}^{s_{2}}=\varepsilon _{1}^{1}\varepsilon _{3}^{3}-\varepsilon _{1}^{2}\varepsilon _{3}^{3}\\C_{e_{1}}^{s_{3}}=-\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{e_{2}}^{s_{3}}=\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{e_{3}}^{s_{2}}=-\varepsilon _{1}^{1}\varepsilon _{2}^{2}&C_{e_{4}}^{s_{2}}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}-\varepsilon _{1}^{1}\varepsilon _{2}^{3}+\varepsilon _{1}^{2}\varepsilon _{2}^{3}\end{array}}} 5297: 4696: 4509: 5292:{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{e_{2}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{e_{3}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}-\varepsilon _{2}^{1}\varepsilon _{1}^{2}\\C_{e_{1}}^{s_{1}}=\varepsilon _{2}^{3}-\varepsilon _{2}^{2}&C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}-\varepsilon _{2}^{1}&C_{e_{3}}^{s_{1}}=\varepsilon _{2}^{2}-\varepsilon _{2}^{1}\\C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}&C_{e_{2}}^{s_{2}}=-\varepsilon _{1}^{1}&C_{e_{3}}^{s_{2}}=\varepsilon _{1}^{1}-\varepsilon _{1}^{2}\\\end{array}}} 7223: 2711: 5573: 5307: 697: 2353: 3094: 1999: 7741: 5318: 2347: 449: 1207: 2706:{\displaystyle {\begin{array}{ll}C_{e_{1}}^{s_{1}}=\varepsilon _{2}^{3}-\varepsilon _{2}^{2}&C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}\\C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}&C_{e_{2}}^{s_{2}}=-\varepsilon _{1}^{1}\\C_{e_{3}}^{s_{1}}=\varepsilon _{2}^{2}&C_{e_{3}}^{s_{2}}=\varepsilon _{1}^{1}-\varepsilon _{1}^{2}\end{array}}} 2877: 2887: 1701: 4667: 7491: 5568:{\displaystyle d=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{4}-\varepsilon _{1}^{1}\varepsilon _{2}^{3}\varepsilon _{3}^{4}-\varepsilon _{3}^{1}\varepsilon _{1}^{2}\varepsilon _{2}^{3}+\varepsilon _{1}^{2}\varepsilon _{2}^{3}\varepsilon _{3}^{4}-\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{3}} 1381: 22:

published the first catalog of control patterns in metabolic control analysis (MCA). His doctoral research. concerned the use of graphical patterns to elucidate chains of interaction in metabolic regulation, later published in the European Journal of Biochemistry. In his thesis, he cataloged 25

2155: 1515: 440: 2127: 692:{\displaystyle \varepsilon _{1}^{2}={\frac {\partial v_{2}}{\partial s_{1}}}{\frac {s_{1}}{v_{2}}},\quad \varepsilon _{2}^{2}={\frac {\partial v_{2}}{\partial s_{2}}}{\frac {s_{2}}{v_{2}}},\quad \varepsilon _{2}^{3}={\frac {\partial v_{3}}{\partial s_{2}}}{\frac {s_{2}}{v_{3}}}} 1018: 1031: 3089:{\displaystyle d=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{3}-\varepsilon _{1}^{1}\varepsilon _{2}^{2}\varepsilon _{3}^{4}+\varepsilon _{1}^{1}\varepsilon _{2}^{3}\varepsilon _{3}^{4}-\varepsilon _{1}^{2}\varepsilon _{2}^{3}\varepsilon _{3}^{4}} 2723: 1994:{\displaystyle {\begin{array}{ll}C_{e_{1}}^{s_{1}}=\varepsilon _{2}^{3}&C_{e_{1}}^{s_{2}}=\varepsilon _{1}^{2}\\C_{e_{2}}^{s_{1}}=-\varepsilon _{2}^{3}&C_{e_{2}}^{s_{2}}=0\\C_{e_{2}}^{s_{2}}=0&C_{e_{3}}^{s_{2}}=-\varepsilon _{1}^{2}\end{array}}} 7736:{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J_{1}}=\varepsilon _{s}^{3}(1-\alpha )+\varepsilon _{s}^{2}\alpha \\C_{e_{1}}^{J_{1}}=-\varepsilon _{s}^{1}\alpha \\C_{e_{1}}^{J_{1}}=-\varepsilon _{s}^{1}(1-\alpha )+\varepsilon _{s}^{2}\alpha \end{array}}} 4520: 802: 1695: 7462: 1215: 7366: 238: 2342:{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=\varepsilon _{1}^{2}\varepsilon _{2}^{3}&C_{e_{2}}^{J}=-\varepsilon _{1}^{1}\varepsilon _{2}^{3}&C_{e_{3}}^{J}=\varepsilon _{1}^{1}\varepsilon _{2}^{2}\\\end{array}}} 1394: 321:. This means that a product elasticity will have identical subscripts and superscripts making them easy to identify. The source boundary species is always labeled zero as well as the label for the first reaction. 329: 2013: 1578: 1202:{\displaystyle C_{v_{1}}^{J}={\frac {\varepsilon _{1}^{2}}{\varepsilon _{1}^{2}-\varepsilon _{1}^{1}}}\qquad C_{v_{2}}^{J}={\frac {-\varepsilon _{1}^{1}}{\varepsilon _{1}^{2}-\varepsilon _{1}^{1}}}} 892: 884: 103: 2872:{\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}S_{2}{\stackrel {v_{3}}{\longrightarrow }}S_{3}{\stackrel {v_{4}}{\longrightarrow }}X_{1}} 58:

are similar in that they describe how a perturbation in a given enzyme affects steady-state flux through a pathway. Such coefficients can be written in terms of elasticity coefficients.

42:

The control equations, especially the numerators of the equations, can give information on the relative importance and routes by which perturbations travel through a biochemical network

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are local properties that describe how a single reaction is influenced by changes in the substrates and products that might influence the rate. For example, given a reaction such as:

4662:{\displaystyle d=\varepsilon _{1}^{1}\varepsilon _{2}^{2}-\varepsilon _{1}^{1}\varepsilon _{2}^{3}+\varepsilon _{1}^{2}\varepsilon _{2}^{3}-\varepsilon _{2}^{1}\varepsilon _{1}^{2}} 7280: 714: 1587: 7375: 1376:{\displaystyle C_{v_{1}}^{s_{1}}={\frac {1}{\varepsilon _{1}^{2}-\varepsilon _{1}^{1}}}\qquad C_{v_{2}}^{s_{1}}={\frac {-1}{\varepsilon _{1}^{2}-\varepsilon _{1}^{1}}}} 293: 39:

toolkit that could generate patterns automatically and symbolically from a description of the network. This software was used to generate the patterns shown below.

7811: 319: 7994: 7484: 5595: 4689: 3116: 2149: 260: 171: 148: 125: 7287: 178: 7866:

Christensen, Carl D; Hofmeyr, Jan-Hendrik S; Rohwer, Johann M (1 January 2018). "PySCeSToolbox: a collection of metabolic pathway analysis tools".

1510:{\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}S_{2}{\stackrel {v_{3}}{\longrightarrow }}X_{1}} 435:{\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}S_{2}{\stackrel {v_{3}}{\longrightarrow }}} 7926:"Delving deeper: Relating the behaviour of a metabolic system to the properties of its components using symbolic metabolic control analysis" 2122:{\displaystyle d=\varepsilon _{1}^{2}\varepsilon _{2}^{3}-\varepsilon _{1}^{1}\varepsilon _{2}^{3}+\varepsilon _{1}^{1}\varepsilon _{2}^{2}} 7771: 55: 51: 50:

Control patterns describe how a perturbation to a given parameter affects the steady-state level of a given variable. For example, a

1530: 242:

To make the notation manageable, a specific numbering scheme is used in the following patterns. If a substrate has an index of

1013:{\displaystyle C_{e_{1}}^{s_{1}}={\frac {1}{\varepsilon _{1}^{2}}}\qquad C_{e_{2}}^{s_{1}}={\frac {-1}{\varepsilon _{1}^{2}}}} 815: 4508: 69: 7766: 7751: 7999: 54:

can describe how the overexpression of a specific enzyme can influence steady-state metabolite concentrations.

7901:

Rohwer, Johann; Akhurst, Timothy; Hofmeyr, Jannie (2008). "Symbolic Control Analysis of Cellular Systems".

7761: 61: 19: 797:{\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}X_{1}} 128: 24: 7232: 3128: 7222: 1690:{\displaystyle {\begin{array}{lll}C_{e_{1}}^{J}=1&C_{e_{2}}^{J}=0&C_{e_{3}}^{J}=0\end{array}}} 7937: 7989: 7827:"Control-pattern analysis of metabolic pathways: Flux and concentration control in linear pathways" 151: 36: 28: 7457:{\displaystyle d=\varepsilon _{s}^{2}\alpha +\varepsilon _{s}^{3}(1-\alpha )-\varepsilon _{s}^{1}} 7906: 7805: 4701: 5306: 7965: 7883: 7848: 7776: 7756: 265: 7955: 7945: 7875: 7838: 23:

patterns for various biochemical networks. In later work, his research group, together with

298: 7941: 5607: 2358: 1706: 7960: 7925: 7843: 7826: 7469: 5580: 4674: 3101: 2134: 245: 156: 133: 110: 7983: 7924:

Christensen, Carl D.; Hofmeyr, Jan-Hendrik S.; Rohwer, Johann M. (28 November 2018).

7879: 7361:{\displaystyle \alpha ={\frac {v_{2}}{v_{1}}}\quad 1-\alpha ={\frac {v_{3}}{v_{1}}}} 7910: 7496: 2160: 1592: 233:{\displaystyle \varepsilon _{s}^{v}={\frac {\partial v}{\partial s}}{\frac {s}{v}}} 7950: 324:

For example, the following fragment of a network illustrates this labeling:

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Studies in steady-state modelling and control analysis of metabolic systems

127:. This reaction rate can be influenced by changes in the concentrations of 7852: 7221: 5305: 4507: 32: 173:. This influence is measured by an elasticity which is defined as: 1582:

Assume that each of the following expressions is divided by d

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