If each gene retains at least one ancestral element, while all redundant elements degenerate, we reach a state known as subfunctionalization. Perhaps the best studied example of this phenomenon involves the paralogous genes Hoxa1 and Hoxb1 in mouse development.
The Strong Collection axiom scheme asserts that if for every x ∈ A there is y such that φ (x, y), then there is a set B such that for every x ∈ A there is y ∈ B such that φ x, y) (as in the usual scheme) but also for every y ∈ B there is x ∈ A such that φ x, y) (B doesn't contain any redundant elements).
Nonetheless, the redundant element comes at a cost, even if it is less than that of developing the first one when both are based on the same design.
Heterogeneous redundancy, i.e., the mixture of redundant element types in each subsystem, is allowed in the formulations.
How, given the many redundancies of metabolic pathways, can interventions be effective before the last redundant element has been irreversibly damaged?
In order to eliminate such redundant elements, a final inter-category key-pose merging process is applied.
For example, in a system with two redundant elements backing each other's function, the removal of any of the two will have no phenotypic effect, leading to the false conclusion that both elements are superfluous.
Functional dissection of Xist RNA has identified a critical element, the A repeat, required for chromosome inactivation, and multiple redundant elements that mediate localization in cis (Wutz et al., 2002).
Given the desired redundancy level and a fixed set of standby mode choices, the objective of the optimal system design is to choose a standby mode for each of redundant elements so as to minimize expected mission cost of the system subject to a certain reliability constraint.
In biological systems, the loss of a component can be compensated by redundant elements (the presence of other identical components) or by degenerate elements (structurally different components that perform the same function with respect to context).
