Department of Molecular Physiology and Biophysics, University of Vermont, Burlington, VT, USA.
Biophys J. 2013 Jun 4;104(11):2540-52. doi: 10.1016/j.bpj.2013.04.045.
Viscoelastic characteristics of many materials falling under the category of soft glassy substances, including biological tissue, often exhibit a mechanical complex modulus Y(ω) well described by a fractional derivative model: Y(ω) = E(iω/ϕ)k, where E = a generalized viscoelastic stiffness; i = (-1)1/2; ω = angular frequency; ϕ = scaling factor; and k = an exponent valued between 0 and 1. The term "fractional derivative" refers to the value of k: when k = 0 the viscoelastic response is purely elastic, and when k = 1 the response is purely viscous. We provide an analytical derivation of the fractional derivative complex modulus based on the hypothesis that the viscoelastic response arises from many intermittent molecular crosslinks, whose lifetimes longer than a critical threshold lifetime, tcrit, are distributed with an inverse power law proportional to t-(k+2). We demonstrate that E is proportional to the number and stiffness of crosslinks formed at any moment; the scaling factor ϕ is equivalent to reciprocal of tcrit; and the relative mean lifetime of the attached crosslinks is inversely proportional to the parameter k. To test whether electrostatic molecular bonds could be responsible for the fractional derivative viscoelasticity, we used chemically skinned human skeletal muscle as a one-dimensional model of a soft glassy substance. A reduction in ionic strength from 175 to 110 mEq resulted in a larger E with no change in k, consistent with a higher probability of interfilament molecular interactions. Thick to thin filament spacing was reduced by applying 4% w/v of the osmolyte Dextran T500, which also resulted in a larger E, indicating a greater probability of crosslink formation in proportion to proximity. A 10°C increase in temperature resulted in an increase in k, which corresponded to a decrease in cross-bridge attachment lifetime expected with higher temperatures. These theoretical and experimental results suggest that the fractional derivative viscoelasticity observed in some biological tissue arises as a mechanical consequence of electrostatic interactions, whose longest lifetimes are distributed with an inverse power law.
许多属于软玻璃物质类别的材料的黏弹性特征，包括生物组织，通常表现出机械复模量 Y(ω)，可以很好地用分数导数模型描述：Y(ω)=E(iω/ϕ)k，其中 E = 广义黏弹性刚度；i = (-1)1/2；ω = 角频率；ϕ = 标度因子；k = 0 到 1 之间的指数值。“分数导数”一词指的是 k 的值：当 k = 0 时，黏弹性响应是纯弹性的，当 k = 1 时，响应是纯粘性的。我们基于黏弹性响应源于许多间歇性分子交联的假设，提供了分数导数复模量的解析推导，其寿命长于临界阈值寿命 t crit 的交联分布具有与 t-(k+2) 成反比的逆幂律。我们证明 E 与任何时刻形成的交联数量和刚度成正比；标度因子ϕ相当于 t crit 的倒数；并且附着交联的相对平均寿命与参数 k 成反比。为了测试静电分子键是否可能是分数导数黏弹性的原因，我们使用化学剥皮的人类骨骼肌作为软玻璃物质的一维模型。从 175 到 110 mEq 的离子强度降低导致 E 增加而 k 不变，这与丝状分子相互作用的可能性更高一致。通过施加 4% w/v 的渗透剂 Dextran T500 来减小厚到薄的丝间距，这也导致 E 增加，表明交联形成的可能性更大，比例接近。温度升高 10°C 导致 k 增加，这对应于较高温度下预期的交联桥接附着寿命缩短。这些理论和实验结果表明，在一些生物组织中观察到的分数导数黏弹性是由于静电相互作用的机械后果，其最长寿命分布具有逆幂律。
