This system is complemented by conservation equations (7) K t = [ K ] + [ SK ], P t = [ P ] + S p P, (8) S t = [ S ] + S p + [ SK ] + S p P, where the total enzyme K t, P t and substrate S t concentrations are in μM and are often assumed to be constants.
Assuming p≪k (this results from the smallness of g), we get to the quadratic Hamiltonian H ex = sk + p 1 2 2 m 1 + p 2 2 2 m 2 − e 2 χr, (3).
It follows from (3) and (9) that αsatisfies (23) α [ SK ] = S p P and hence, (24) [ SK ] < P t α for P t > 0. The asymptotic expansion of [ SK] with respect to powers of 1 K t in (22) implies that the asymptotic limit of [ SK], i.e., S K ∞ K t = lim K t → ∞ [ SK ] satisfies the quadratic equation (S K ∞ K t ) 2 − u 1 S K ∞ K t − u 2 = 0.
Subject to P SK ≤P (resp. P KD ≤P) and P SK Z 0≤I (resp. P KD Z k ≤I), then, the optimal power (P_{textit {IS}}^{k}) (resp. (P_{textit {IR}}^{k})) and (P_{textit {US}}^{k}) (resp. (P_{textit {UR}}^{k})) used by CS (resp. CR k ) in interweave and underlay respectively are as follow: P_{IS}^{k} ;= P_{IR}^{k} = P (4).
In the interweave approach, there is just the maximum transmission power constraint, then, P SK ≤P and P KD ≤P.
In the underlay approach, the CS's (resp. CR k ) transmission power is constrained as P SK Z 0≤I (resp. P KD Z k ≤I), where I is the interference threshold that the interference generates by CS (resp. CR k ) on the PD remains below it, and is constrained as P SK ≤P (resp. P KD ≤P), where P is the maximum transmission power.
□ Corollary 2. If P t α < K t, then [ SK ] → P t α and R → 1 as S t → ∞.
If P t α < K t, then [ SK ] → P t α and R → 1 as S t → ∞, otherwise, [ SK] → K t and R → 0 as S t → ∞.
