This question is extremely broad because, rather than there being a ‘one-click’ answer, whole areas of amino acid metabolism need to be considered. This is both inappropriate and impossible to do here — even if I understood everthing about it (which I certainly don’t). However I shall provide a specific biochemical answer (or partial answer, as all is not understood) regarding the regulation of the urea cycle, after commenting on some of the assumptions implicit in the question.
Comments on the assumptions of the question
I am not sure that ‘protein deficit’ is a useful metabolic construct. From the point of view of a clinician, yes, one can say that someone is suffering from protein deficiency or malnutrition, but in metabolic terms protein is a not the highest priority and will be broken down to provide glucose for the brain during long-term starvation. And it is well established that protein synthesis is under the control of hormones and growth factors (reviewed here), and this will affect the fate of dietary amino acids or those produced by protein turnover. In addition, the emphasis on hormones in the question seems to ignore the role of the concentration of amino acids and other intermediates in regulating metabolic pathways. Finally, there is control of the expression of the genes encoding enzymes of the urea cycle, which operates over a longer time period.
The Medical Biochemistry site provides a wider perspective on amino acid metabolism and nitrogen metabolism and the urea cycle.
Regulation of the Urea Cycle
The first point that should be emphasized is that the urea cycle does not exist in isolation from other metabolic pathways, but is intimately connected to them. This is shown in the diagram below:
It should be remembered that ammonia for the urea cycle only comes from the deamination of glutamate by glutamate dehydrogenase, and the transamination of other amino acids to produce this depends on a pool of α-ketoglutarate (2-oxo glutarate). A requirement for α-ketoglutarate also exists in relation to the regeneration of aspartate for the cycle itself.
The entry point of ammonia into the urea cycle is the reaction catalysed by carbamoyl phosphate synthetase:
It turns out that this reaction requires N-acetyl glutamate (NAG) for activity, and it is thought that this plays a regulatory role in the urea cycle, as increased intake of protein has been found to increase the hepatic content of NAG. Although aspects of this remain to be clarified or resolved, it is thought that the regulation occurs at the point at which NAG is synthesized from glutamate in a reaction catalysed by the enzyme N-acetyl glutamate synthase:
This enzyme is activated by arginine, giving rise to a model for regulation described more fully in a review by Caldovic and Tuchman, from which I quote the following:
“… the hypothesis that NAG is a regulator of ureagenesis… If this hypothesis is correct, mitochondrial concentrations of glutamate and/or arginine may reflect the need for nitrogen disposal through their effect on the production of NAG by NAG synthase, glutamate as a substrate and arginine as an effector.”