三維細胞抓力之量測與數值分析

Abstract

在細胞生物學領域中，了解細胞如何施力以及反應外界的環境是相當重要 且關鍵的，當細胞在移動時，細胞會透過複雜的機制對附著的基材產生抓力，為使我們能夠更了解細胞究竟如何與細胞外基質作用，測量與分析細胞抓力的方法佔了重要的一席之地。本研究採用將細胞培養在以聚丙烯醯胺凝膠（polyacrylamide gel，簡稱PA gel）製備而成的基材上，測量與分析其細胞抓力。聚丙烯醯胺凝膠為黏彈性材料，但若考慮黏彈性的材料性質會使得測量細胞抓力的情況過為複雜，因此目前在此相關領域中皆是以材料性質為線性彈性的假設下進行分析。針對細胞培養在平面基材上的實驗設置下，有相當多種分析細胞抓力的方法，傳統的方法以化簡為二維分析為主，忽略在垂直於基材表面方向所造成的位移和抓力，且普遍以由三維的包辛尼斯克-塞魯蒂方程式所化簡推導而得的二維格林函數為基礎作運算，其中又以傅立葉轉換法最為廣泛使用。然而近年的文獻多指出當細胞黏附在平面的基材上時，會對細胞外基質同時施加剪切和垂直的抓力，因此細胞在垂直方向的影響是不可忽略的，這也是近年來建立三維細胞抓力分析方法的重要性。與傅立葉轉換法相比，有限元素法能夠有效的應用於二維、三維抓力分析，並且能推廣至有複雜幾何形狀的基材和非彈性材料，不受限於包辛尼斯克-塞魯蒂方程式的限制。 本研究主要著重在計算方法的部分，並建立了各種分析方法以探討不同方 法間的差異。實驗中是利用追蹤細胞造成的內部螢光小球的位移場而回推細胞抓力，因此為了解基材中螢光小球濃度的影響，本研究利用模擬不同濃度的資料點密度，歸納出抓力能夠被回復的範圍。除此之外，我們也發現螢光小球密度較低時，傾向於更加低估抓力的大小值，且也導致回復的抓力位置有誤差。另外，本研究也藉由模擬分析細胞抓力回復和應變能回復來探討實驗雜訊所造成的影響，並且測試不同方法對雜訊影響的穩定度。透過自助抽樣法的結果， 實驗雜訊確實會增加細胞抓力偏差的標準差。而在二維分析方法中，在剪切抓力方向造成的標準差均約為20 帕，因此我們認為傅立葉分析法與二維的有限元素法對於雜訊處理的穩定度是足夠的。除了三維分析方法對於細胞在垂直方向的抓力計算之外，由於二維分析方法忽略垂直基材方向的影響，導致大量低估細胞的應變能，因此這也是提高三維分析方法重要性的原因之一。綜合而言，本研究建立了不同的細胞抓力分析方法，並比較各種方法間的差異，以及利用模擬方法提供細胞實驗部分建議的條件以及有用的資訊。

While migrating, cells generate traction forces to the polyacrylamide substrate through complex adhesion mechanism. Measuring such forces, known as traction force microscopy, is of critical importance in understanding how cells interact with extracellular matrix. The polyacrylamide gel is indeed a viscoelastic material. However, cellular traction force analysis considering viscoelastic models becomes too complicated to tackle, and is beyond the scope of this study. In this thesis, we focus on the traction force under static condition; therefore, we assume that the gel is linearly elastic material and conduct the numerical analysis accordingly. Some analytical methods based on Boussinesq-Cerruti problem [1], such as two-dimensional Fourier-transform traction cytometry (FTTC) [2], have been widely used to reconstruct the cellular traction from substrate displacement field measured by confocal microscopy, and they often assume that cells exert only shear forces on an elastic flat substrate. However, recent studies indicated that cells on a planar substrate exert significant out-of-plane traction. Therefore, the out-of-plane component of the traction should not be neglected, and we found that neglecting the out-of-plane component results in considerable deviation from the more realistic traction force. Compared with the conventional FTTC method, finite element method (FEM) can be readily applied to substrates with complicated non-planar surfaces and those made of inelastic materials. In this thesis, we focus on the computational methods, and develop several kinds of methods to understand the difference between different methods. In order to understand the influences of beads density, which represent the quantity of data points, we use simulations to find resolvable range for single focal adhesion recovery under different beads density. In addition, we also investigate whole normal traction field recovery and find that the lower beads density causes the recovered traction more dispersed and underestimated, and that also probably causes the deviation on the recovered positions of tractions. While further adding the noises effects during the traction force reconstruction, we investigate the strain energy recovery and traction force recovery, and test the reliability of different methods for noises treatment. We find that noises increasing the traction deviations between bootstrap iterations. For 2D analysis methods, there are only about 20 Pa of standard deviations in shear tractions, thus we think both the FTTC and 2D FEM are consistent enough for noises treatment. Beside the analysis of normal tractions, due to the neglecting in z direction, 2D analysis methods tend to much more underestimate the strain energy than the recovered strain energy in 3D analysis, and this also confirm the significance for 3D methods. In conclusion, we develop several analysis methods and compare between different methods, and use various simulations to provide the direction for improvement and helpful information for a given experimental setup.