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Hill is an uninstantiated function used to indicate parameter values for a hill-function based reaction.

Hill[..] and hill[..] may be used interchangeably.

## Catalytic Hill-Function Reactions

The canonical form for a **catalytic hill reaction** is:

The interpreted differential equation is:

In a catalytic hill reaction both the reactant (A) and the product (B) change values. The catalyst (En) does not change. The first term here produces an inhibitory Hill function, while the second one produces an activating Hill function. To get only inhibition, set v=0; to get only activation, set v1=0.

All parameters are optional and may be omitted starting from the right. Default parameter values are: v=1, n=1, K=1, β=1, T=1, v1=0:

hill[v,n,k,β,T] is equivalent to hill[v, n, k, β, T, 0]

hill[v,n,k,β] is equivalent to hill[v, n, k, β, 1, 0]

hill[v,n,k]] is equivalent to hill[v, n, k, 0, 1, 0]

hill[v,n] is equivalent to hill[v, n,1, 0, 1, 0]

hill[v] is equivalent to hill[v, 1, 1, 0, 1, 0]

hill[] is equivalent to hill[1, 1, 1, 0, 1, 0]

### Multiple Substrates - Equivalent Forms

If multiple catalytic reactions have the same product their corresponding ODE terms are added. Note that this is different from the way regulatory hill reactions are combined:

## Regulatory Hill Reactions

The canonical form for a **regulatory hill reaction** is:

In a regulatory hill reaction, the value of the product (B) changes but the value of the regulator (A) does not:

The first term in the equation for B here is the inhibitory hill function, and the second term is the activating. To get only an activating function, set v1 to 0; to get only an inhibitory function, set v to 0.

If there are multiple regulatory hill reactions with the same product, then they are combined vectorally using the T's as weights:

Note that all parameters except for (T) are ignored in all but the first reaction.

The following is equivalent: