Gatenby Robert, Frieden B Roy
Departments of Integrated Mathematical Oncology and Radiology, Moffitt Cancer Center, Tampa, FL 33612, United States of America.
College of Optical Sciences, University of Arizona, Tucson, AZ 85721, United States of America.
PLoS One. 2016 May 5;11(5):e0154867. doi: 10.1371/journal.pone.0154867. eCollection 2016.
Enzymes are proteins that accelerate intracellular chemical reactions often by factors of 105-1012s-1. We propose the structure and function of enzymes represent the thermodynamic expression of heritable information encoded in DNA with post-translational modifications that reflect intra- and extra-cellular environmental inputs. The 3 dimensional shape of the protein, determined by the genetically-specified amino acid sequence and post translational modifications, permits geometric interactions with substrate molecules traditionally described by the key-lock best fit model. Here we apply Kullback-Leibler (K-L) divergence as metric of this geometric "fit" and the information content of the interactions. When the K-L 'distance' between interspersed substrate pn and enzyme rn positions is minimized, the information state, reaction probability, and reaction rate are maximized. The latter obeys the Arrhenius equation, which we show can be derived from the geometrical principle of minimum K-L distance. The derivation is first limited to optimum substrate positions for fixed sets of enzyme positions. However, maximally improving the key/lock fit, called 'induced fit,' requires both sets of positions to be varied optimally. We demonstrate this permits and is maximally efficient if the key and lock particles pn, rn are quantum entangled because the level of entanglement obeys the same minimized value of the Kullback-Leibler distance that occurs when all pn ≈ rn. This implies interchanges pn ⇄ brn randomly taking place during a reaction successively improves key/lock fits, reducing the activation energy Ea and increasing the reaction rate k. Our results demonstrate the summation of heritable and environmental information that determines the enzyme spatial configuration, by decreasing the K-L divergence, is converted to thermodynamic work by reducing Ea and increasing k of intracellular reactions. Macroscopically, enzyme information increases the order in living systems, similar to the Maxwell demon gedanken, by selectively accelerating specific reaction thus generating both spatial and temporal concentration gradients.
酶是蛋白质,它们通常能将细胞内化学反应的速度提高10^5 - 10^12倍。我们提出,酶的结构和功能代表了DNA中编码的可遗传信息的热力学表达,其翻译后修饰反映了细胞内和细胞外的环境输入。蛋白质的三维形状由基因指定的氨基酸序列和翻译后修饰决定,允许与底物分子进行几何相互作用,传统上用钥匙-锁最佳匹配模型来描述。在这里,我们应用库尔贝克-莱布勒(K-L)散度作为这种几何“匹配”和相互作用信息含量的度量。当散布的底物pn和酶rn位置之间的K-L“距离”最小时,信息状态、反应概率和反应速率最大。后者服从阿伦尼乌斯方程,我们表明该方程可以从最小K-L距离的几何原理推导出来。推导首先限于固定酶位置集的最佳底物位置。然而,最大程度地改善钥匙/锁匹配,即所谓的“诱导契合”,需要两组位置都进行最佳变化。我们证明,如果钥匙和锁粒子pn、rn是量子纠缠的,这是可行的且效率最高,因为纠缠程度服从当所有pn≈rn时出现的相同最小化库尔贝克-莱布勒距离值。这意味着在反应过程中随机发生的pn⇄brn互换会相继改善钥匙/锁匹配,降低活化能Ea并提高反应速率k。我们的结果表明,通过降低K-L散度来确定酶空间构型的可遗传信息和环境信息的总和,通过降低Ea和提高细胞内反应的k,转化为热力学功。从宏观上看,酶信息增加了生命系统的有序性,类似于麦克斯韦妖思想实验,通过选择性地加速特定反应从而产生空间和时间浓度梯度。