Feasible sign patterns

Identifying constraints on reaction directions is essential for applications of metabolic flux analysis. However, in many applications the procedure used for determining reaction directions is not concretely defined. Typically, a subset of the reactions in a model is assigned as irreversible and the feasible directions are assigned based on information in pathway databases [59], In these applications, by treating certain reactions as implicitly unidirectional, biologically reasonable results can often be obtained without considering the system thermodynamics as outlined above. 

As an alternative to ad hoc procedures for assigning reaction directions, it is possible to determine reactions directions from first principles based on the thermodynamic constraint defined in the preceding section and knowledge of the direction of the transport flux directions. In fact, it is possible to mathematically state the thermodynamic constraint of Equations (9.20) and (9.21) in an alternative form in terms of the sign pattern of the vector J [16]. 

To define the thermodynamic constraint in terms of sign patterns, it is first necessary to define the concept of a sign vector. A sign vector is a vector with possible entries 0, +, and —. The operation sign ( ) is defined to return the sign vector associated with a vector of real numbers. For example sign —0.1, +5, 0, —2.1 = — + 0, —.  

it is necessary to define the concept of orthogonality of sign vectors. Two sign vectors a and b are said to be orthogonal (a T b) if either (1) the supports of a and b have no indices in common, or (2) there is an index i for which a, and bi have the same signs and there is another index j (j i) for which aj and bj have opposite signs. Given these definitions, the thermodynamic constraint may be stated as  

For a stoichiometric matrix associated with the internal reactions of a given system, with right null space R, the vector of internal fluxes J is thermodynamically feasible if and only if 

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Figure 9.7 Enumeration of all feasible flux sign patterns for example from Figure 9.6. Figure adapted from [216],...

The first pattern in Figure 9.8 is disqualified because there is no way to achieve steady state balance of mass of A or D given this sign pattern. The second pattern is disqualified because there is no feasible chemical potential pattern associated with this flux pattern. 

Therefore the above sign pattern is not thermodynamically feasible because it is not orthogonal to the sign pattern of a vector from the right null space of the stoichiometric matrix of internal reactions. 

Is the flux sign pattern in exercise 34 thermodynamically feasible Would it be thermodynamically feasible for reaction 4 to operate in the reverse direction Why or why not ... 

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