# 希爾方程 (生物化學)

Biochemical binding curves showing the characteristically sigmoidal curves generated by using the Hill equation to model cooperative binding. Each curve corresponds to a different Hill coefficient, labeled to the curve's right. The vertical axis displays the fraction of occupied ligand-binding sites on a protein receptor (${\displaystyle \theta }$), equal to ratio of the concentration of ligand-bound protein ([${\displaystyle \left[PL_{n}\right]}$) to the total concentration of protein receptor (${\displaystyle \left[P_{\text{Total}}\right]}$). The horizontal axis is the ratio of the ligand concentration producing half occupation (${\displaystyle K_{A}}$) to the free ligand concentration (${\displaystyle \left[L\right]}$).

${\displaystyle \theta ={[L]^{n} \over K_{d}+[L]^{n}}={[L]^{n} \over (K_{A})^{n}+[L]^{n}}}$

${\displaystyle \theta }$ - 占用位點的分數，在占用位點處配體可以結合到受體蛋白的活性位點

${\displaystyle [L]}$ - 游離的（未結合的）配體濃度

${\displaystyle K_{d}}$ - 表觀解離常數來源於質量作用定律（對於解離的平衡常數）

${\displaystyle K_{A}}$ - 產生半數占用時的配體濃度（配體濃度足以占用結合位點的一半數目），亦為微觀的解離常數

${\displaystyle n}$ - 希爾係數，描述了協同性（或亦可能是其他生物化學性質，取決於使用希爾方程時的討論背景）

${\displaystyle \log \left({\theta \over 1-\theta }\right)=n\log {[L]}-\log {K_{d}}.}$

• ${\displaystyle n>1}$ - 正協同反應：一旦一個配體分子結合到酶上，酶對其他配體的親和力就會增加。
• ${\displaystyle n<1}$ - 負協同反應：一旦一個配體分子結合到酶上，酶對其他配體的親和力就會減小。
• ${\displaystyle n=1}$ - 非協同反應：酶對於一個配體分子的親和力並不取決於是否有配體分子已結合到其上。

Endrenyi