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- Abstract
- 1 Introduction
- 2 Materials and Methods
- 3 Biomolecular Simulation
- 4 Results
- 5 Discussion
- References

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J Nanotechnol Eng Med. Author manuscript; available in PMC 2010 December 7.

Published in final edited form as:

J Nanotechnol Eng Med. 2010 August 1; 1(3): 10.1115/1.4001934.

doi: 10.1115/1.4001934PMCID: PMC2998284

NIHMSID: NIHMS230338

Asit K. Saha, Center for Allaying Health Disparities through Research and Education (CADRE), Department of Mathematics and Computer Science, Central State University, Wilberforce, OH 45384;

Asit K. Saha: ude.etatslartnec@ahasa; Sean S. Kohles: ude.xdp.scec@selhok

See other articles in PMC that cite the published article.

Understanding physicochemical interactions during biokinetic regulation will be critical for the creation of relevant nanotechnology supporting cellular and molecular engineering. The impact of nanoscale influences in medicine and biology can be explored in detail through mathematical models as an in silico testbed. In a recent single-cell biomechanical analysis, the cytoskeletal strain response due to fluid-induced stresses was characterized (Wilson, Z. D., and Kohles, S. S., 2010, “Two-Dimensional Modeling of Nanomechanical Strains in Healthy and Diseased Single-Cells During Microfluidic Stress Applications,” J. Nanotech. Eng. Med., **1**(2), p. 021005). Results described a microfluidic environment having controlled nanometer and piconewton resolution for explorations of multiscale mechanobiology. In the present study, we constructed a mathematical model exploring the nanoscale biomolecular response to that controlled microenvironment. We introduce mechanical stimuli and scaling factor terms as specific input values for regulating a cartilage molecule synthesis. Iterative model results for this initial multiscale static load application have identified a transition threshold load level from which the mechanical input causes a shift from a catabolic state to an anabolic state. Modeled molecule homeostatic levels appear to be dependent upon the mechanical stimulus as reflected experimentally. This work provides a specific mathematical framework from which to explore biokinetic regulation. Further incorporation of nanomechanical stresses and strains into biokinetic models will ultimately lead to refined mechanotransduction relationships at the cellular and molecular levels.

Understanding the structure, function, and regulation of biological systems in real time at the nanoscale level is a challenging task. Nanobioscience is a newly emerging field where investigators describe the smallest molecular structures and their dynamics in the realm of space and time. Clarifying the functions of cells and biomolecules within the extracellular space will provide the basis for the development of nanoscale devices mimicking and/or influencing biological processes. Knowledge about the topology of the cells, biomolecules, and their mutual interactions at the nanoscale will provide insight regarding fundamental biological functions, which are absolutely necessary in the development of new tissues and/or controlling tissue remodeling.

At the nanoscale level, physiologic regulation of cartilage biomolecules is maintained by chondrocytes although they are sparsely distributed within the tissue and are low in number [1–3]. Cell-cell and cell-extracellular matrix (ECM) interactions and subsequent communication are very important in tissue development, remodeling, and homeostasis. In normal or pathological situations, cartilage ECM homeostasis is dependent upon nanoscale anabolic and catabolic pathways controlled by the chondrocytes [4,5]. Ironically, chondrocytes have dual roles; they produce both the structural macromolecules in tissue formation, such as collagen and proteoglycans, as well as a variety of metalloproteinases, such as collagenase, gelatinase, stromelysin, and aggrecanase, which are responsible for tissue turnover (Fig. 1). Structural protein molecules consist of type II collagen and elastin. Specialized protein molecules consist of fibrillin, fibronectin, laminin, and, lastly, proteoglycans, which are core protein molecules to which long chains of repeating disaccharide units known as glycosaminoglycans (GAG) are attached. The mechanical behavior of cartilage is strongly influenced by constituent content and the electrostatic forces between matrix charge groups [6].

During healthy balanced regulation, the growth factors such as transforming growth factor-*β* (TGF-*β*), insulin-like growth factor-1 (IGF-1), and osteogenic protein-1 (OP-1) stimulate the chondrocytes to synthesize the structural macromolecules, such as collagen and proteoglycan (anabolic modulation), while cytokines interleukin-1 (IL-1), interleukin-6 (IL-6), and tumor necrosis factor-*α* (TNF-*α*) stimulate chondrocyte secretion of proteinases, causing ECM degradation (catabolic modulation). It is now obvious that growth, maintenance, and degradation of cartilage ECM are dependent on a balance between molecular (nanoscale) anabolic and catabolic activities. In the case of cartilage matrix growth and repair/remodel, anabolic processes exceed the catabolic activities. Alternatively, catabolic processes exceed anabolic activities when disease and degeneration are at hand. These relationships have been observed in controlled osteoarthritic culture models where experimental results show that the combination of anabolic growth factors and protective catabolic blockers may be a means for partial restoration of the cartilage matrix [7]. Therefore, any alterations in the molecular regulatory balance between anabolic and catabolic signaling can lead to multiscale pathology (tissue, joint, etc.) including osteoarthritis [8].

The question arises as to whether growth factors and cytokines are solely responsible for maintaining the balance between the anabolic and catabolic regulatory mechanisms, or is some other mechanism involved in this complex process. It has been observed that physical movements and applied mechanical forces maintain healthy cartilage tissue [9,10]. The structural organizations of collagen and proteoglycan in the articular cartilage vary along the depth of the tissue influencing the load-bearing capacity of the tissue. A reduction in the thickness of the cartilage tissue has been observed during an absence of mechanical loading [10]. In the articular cartilage, the concentration of lubricating molecules (such as lubricin) is high around the surface, whereas higher concentrations of aggrecans are found in deeper zones [11,12]. Applied mechanical compression results in a deformity of the cartilage structural molecules while subsequent unloading causes the influx of nutrients from the surrounding synovial fluid into the tissue [13–15]. However, the mechanical influence on the balanced regulatory molecular network of anabolic and catabolic pathways has not been fully described, including the maintenance of ECM homeostasis at the nanoscale level [8].

It has been observed in both in vivo and in vitro environments that injurious loading caused by high mechanical compression of cartilage results in cell apoptosis [16,17] as well as increased degradation and decreased synthesis of matrix structural molecules [18–21]. A similar scenario was observed in tissue engineered constructs where dynamic mechanical stresses influenced the biosynthesis of matrix molecules and accumulation while concurrently reducing the intensity of the catabolic process [22–27]. Alternatively, static loads inhibited the synthesis of matrix biomolecules [22,28].

In our previous predictions of articular cartilage matrix synthesis, an approach was based on systems biology. Our results clearly indicate that anabolic actions of different growth factors are essentially dose and time dependent [29,30]. These works have culminated in characterizing molecular regulatory mechanisms influencing both anabolic and catabolic processes, which maintain matrix homeostasis [31]. With advances in single-cell biomechanics and nanomechanical modeling, the dynamics of anabolic and catabolic pathways can now be extended to include mechanical stimuli, be it static or dynamic [32,33]. The objective of this study is to link the complex dynamics of the molecular anabolic and catabolic pathways with nanomechanical stimuli at the single-cell level. This mathematical modeling approach comes from a systems point of view to help explain the multiscale feedback effects of the anabolic and catabolic actions within an engineered cartilage platform.

Based on previous studies, it can be concluded that there exists a dynamic relationship between anabolic and catabolic activities, which regulates matrix homeostasis. The important anabolic proteins are various growth factors, and the important catabolic protein includes cytokines. During healthy balanced regulation, growth factors such as TGF-*β*, IGF-1, and OP-1 stimulate the chondrocytes to synthesize the structural macromolecules, while cytokines such as IL-1, IL-6, and TNF-*α* stimulate chondrocyte secretion of proteinases, causing ECM degradation.

For the mathematical model, let [*C*], [*G*], and [*E*] represent the nanoscale concentration of biomolecules representing cytokines, growth factors, and ECM, at any particular time, *t*. According to the schematic relationships (Fig. 1), the mathematical model is defined as follows:

$$\begin{array}{c}\frac{d[C]}{dt}={\eta}_{1}[C]\{{\Re}_{2}-[G]\}-{\mu}_{1}[C]\\ \frac{d[G]}{dt}={\eta}_{2}[G]\{[C]-{\Re}_{1}\}-{\mu}_{2}[G]\\ \frac{d[E]}{dt}=\frac{v[G]}{K+[C]}-\sigma [E]\end{array}$$

(1)

where *η _{i}* denotes the overall growth rates,

The dimensionless form of the modeled system can be written as

$$\begin{array}{c}\frac{d}{dT}\left(\begin{array}{c}C\\ G\end{array}\right)=\left(\begin{array}{c}{f}_{1}(C,G,{\alpha}_{1},{\beta}_{1},{\Omega}_{2})\\ {f}_{2}(C,G,{\alpha}_{2},{\beta}_{2},{\Omega}_{1})\end{array}\right)\\ \frac{d}{dT}E={f}_{3}(C,G,E,\lambda ,\delta )\end{array}$$

(2)

where

$$\begin{array}{llll}\phantom{\rule{1em}{0ex}}C={K}^{-1}[C],& G={K}^{-1}[G],& E=K[E],& T={K}^{2}[t]\\ {\alpha}_{1}={K}^{-1}{\eta}_{1},& {\alpha}_{2}={K}^{-1}{\eta}_{2},& {\beta}_{1}={K}^{-2}{\mu}_{1},& {\beta}_{2}={K}^{-2}{\mu}_{2}\\ \phantom{\rule{1em}{0ex}}\lambda ={K}^{-1}v,& \delta ={K}^{-2}\sigma ,& {\Omega}_{1}={K}^{-1}{\Re}_{1},& {\Omega}_{2}={K}^{-1}{\Re}_{2}\end{array}$$

(3)

and

$$\begin{array}{c}{f}_{1}(C,G,{\alpha}_{1},{\beta}_{1},{\Omega}_{2})={\alpha}_{1}C({\Omega}_{2}-G)-{\beta}_{1}C\\ {f}_{2}(C,G,{\alpha}_{2},{\beta}_{2},{\Omega}_{1})={\alpha}_{2}G(C-{\Omega}_{1})-{\beta}_{2}C\\ {f}_{3}(C,G,E,\lambda ,\delta )=\frac{\lambda G}{1+C}-\delta E\end{array}$$

(4)

Here, the first equation in system (2) describes both the anabolic and catabolic actions of cytokines and growth factors, whereas the second equation demonstrates the synthesis dynamics of the ECM.

The kinetic conditions are now divided into two environments, with and without applied mechanical stresses. In the mechanical stress environment such as experienced in vivo, the system experiences a constant mechanical stimulus. In this mechanical loading system, Eq. (2) can be written as

$$\begin{array}{c}\frac{d}{dT}\left(\begin{array}{c}C\\ G\end{array}\right)=\left(\begin{array}{c}{f}_{1}(C,G,{\alpha}_{1},{\beta}_{1},{\Omega}_{2})\\ {f}_{2}(C,G,{\alpha}_{2},{\beta}_{2},{\Omega}_{1})\end{array}\right)+\left(\begin{array}{cc}{\rho}_{1}& 0\\ 0& {\rho}_{2}\end{array}\right)\left(\begin{array}{c}{\xi}_{1}(T)\\ {\xi}_{2}(T)\end{array}\right)\\ \frac{d}{dT}E={f}_{3}(C,G,E,\lambda ,\delta )\end{array}$$

(5)

where $\left(\begin{array}{c}{\xi}_{1}(T)\\ {\xi}_{2}(T)\end{array}\right)$ represents the applied mechanical stimuli and $\left(\begin{array}{cc}{\rho}_{1}& 0\\ 0& {\rho}_{2}\end{array}\right)$ denotes scaling factor coefficients associated with this influence on cytokine (C) and growth factor (G) dynamics.

In a mechanically protected system where a physical “shielding” and no mechanical loading occurs, the nontrivial steady state and/or equilibrium levels of the constituents (*C*_{ss},*G*_{ss},*E*_{ss}) can be obtained from the following set of equations [31]:

$$\left(\begin{array}{c}{C}_{\text{ss}}\\ {G}_{\text{ss}}\\ {E}_{\text{ss}}\end{array}\right)=\left(\begin{array}{c}{\Delta}_{1}\\ {\Delta}_{2}\\ \frac{\lambda {\Delta}_{2}}{\delta (1+{\Delta}_{1})}\end{array}\right)$$

(6)

where Δ_{1} = Ω_{1} +*β*_{2}/*α*_{2} and Δ_{2} = Ω_{2}−*β*_{1}/*α*_{1}.

The two main biomolecular constituents of ECM considered here are type II collage, the structural proteins, and proteoglycans. Proteoglycans are the core proteins to which are attached long chains of repeating disaccharide units known as GAG. As shown previously for steady state ECM,

$${E}_{\text{ss}}\approx {\text{Collagen}}_{\text{ss}}+{\text{GAG}}_{\text{ss}}=\frac{\lambda \left({\Omega}_{2}-\frac{{\beta}_{1}}{{\alpha}_{1}}\right)}{\delta \left(1+{\Omega}_{1}+\frac{{\beta}_{2}}{{\alpha}_{2}}\right)}$$

(7)

If we assume that the synthesis of collagen and GAG molecules are independent, then we can separate the ECM components and rewrite the system in a fashion similar to Eq. (2), such as

$$\frac{d}{dT}\left(\begin{array}{c}\text{GAG}\\ \text{Collagen}\end{array}\right)=\left(\begin{array}{c}\frac{{\lambda}_{1}G}{1+C}-{\delta}_{1}(\text{GAG})\\ \frac{{\lambda}_{2}G}{1+C}-{\delta}_{2}(\text{Collagen})\end{array}\right)$$

(8)

The steady state conditions of the GAG and collagen components can then be defined as

$$\left(\begin{array}{c}{\text{GAG}}_{\text{ss}}\\ {\text{Collagen}}_{\text{ss}}\end{array}\right)=\left(\begin{array}{c}\frac{{\lambda}_{1}\left({\Omega}_{2}-\frac{{\beta}_{1}}{{\alpha}_{1}}\right)}{\sum}\\ \frac{{\lambda}_{2}\left({\Omega}_{2}-\frac{{\beta}_{1}}{{\alpha}_{1}}\right)}{\sum}\end{array}\right)$$

(9)

where

$$\sum =\delta \left(1+{\Omega}_{1}+\frac{{\beta}_{2}}{{\alpha}_{2}}\right)>0$$

A phase-space diagram of this dynamic system indicates accumulation-dependent zones delineated by an abscissa representing cytokine levels and an ordinate indicating growth factor status, intersecting at a homeostatic origin (*C*_{ss},*G*_{ss}) [31].

Normalized parameter estimation associated with anabolic/catabolic activity was made through a mathematical association between growth factor/cytokine abundance in the development of sweat glands in human fetal skin [34,31]. These input parameters provided the synergistic effects of cytokines and growth factors over a time frame typical of recent engineered cartilage culture scenarios [32].

The overall parameter estimations include biokinetic and biomechanical factors (Tables 1 and and2).2). The parameters characterizing ECM structural molecule and proteoglycan dynamics were determined previously through a deterministic approach where kinetic rate ratios of synthesis per decay contribute to steady state levels of ECM accumulation [30]. The parameters associated with mechanical stress and strain within isolated cartilage cells were also derived from previous analyses. These works determined the experimental and theoretical stresses applied to suspended cell-sized polystyrene microspheres within microfluidic environments designed for biological cell investigations. The stress analysis showed that fluid-induced stresses on suspended cells ranged from 0.02 Pa to 0.04 Pa during physically limited in vitro experiments [32]. However, the numerical strain analysis indicated the required applied stresses nearing 10.0 Pa to elicit an appreciable strain response depending on the health (and stiffness) of the cell [33].

Iterative analyses of the described biokinetic models were run through 5000 normalized time steps using commercial software (mathcad 14.0, Parametric Technology Corp., Needham, MA). A range of nano- to mesomechanical stimuli identified in the previous single-cell studies was included in the time modeling iterations through the normalized parameters, *ξ*_{i} and *ρ*_{i}, introduced in Eq. (5). For this study, the load scaling factors were held constant at *ρ*_{1}(*T*) = *ρ _{2}*(

A mathematical framework has been presented, which characterizes nanoscale biokinetic mechanisms as stimulated within the micromechanical environment of a single cell. The overall impact of the mechanical stress has influenced the two fundamental regulatory processes, namely, anabolic and catabolic pathways, which lead to matrix homeostasis. The time-dependent GAG and collagen accumulation is shown in dimensionless form where the mechanical stress lies in the interval 0.0≤*ξ*(*T*)≤1.0 with scaling factors *ρ*_{1} = *ρ*_{2}=1 (Figs. 2–4). The imposed mechanical stress is considered here as a constant function over time; i.e., it is represented as a static mechanical stimulus. Three different experimental data sets have been considered [36–38].

Comparing the results with input data from previous work [36], the static mechanical stress in the range of 0.0≤*ξ*(*T*)≤1.0 may have little influence on GAG accumulation toward matrix homeostasis (Figs. 2(a)–2(c)). However, the collagen accumulation attains peak values within a very short period of time when there is no mechanical stress. One can assume from a system dynamics point of view this peak value as a “pseudo-steady-state,” which is unstable. From the simulation, the system attained this pseudo-steady-state around the 2000th dimensionless time step. If there is no mechanical stress applied, the collagen accumulation eventually attains a permanent steady state at a much lower value, at least 50% less than its peak value, in a longer time frame (Fig. 2(a)). A threshold biokinetic response was observed in the model as the mechanical stress state was increased in scale (Figs. 2(b) and 2(c)), although GAG accumulation was minimally affected by the stimulation.

From an alternative data set [37], an intrinsic rate increase (growth rate minus decay rate) of GAG molecules was observed to be much higher than that of collagen. As a consequence, GAG accumulation was much higher than collagen accumulation (Figs. 3(a)–3(c)). However, it appears that collagen accumulation was less sensitive to lower levels of mechanical stimuli (*ξ*≤0.02) than GAG, requiring greater stimulation to reach homeostatis. In a third data set [38], the amount of accumulation between the GAG and collagen molecules also differed (Figs. 4(a)–4(c)). However, their kinetic sensitivity to mechanical stimulation appeared similar with a higher homeostatic threshold (*ξ*≤1.0).

The phase-space dynamics of the anabolic and catabolic pathways under an increasing influence of mechanical stimuli, 1.0≤*ξ*(*T*)≤100.0, is demonstrated (Fig. 5). In a previous work [31], it was observed that the matrix molecules attain an apparent homeostasis within a very short period of time without any mechanical stimuli. The oscillatory dynamics between anabolic and catabolic interactions is shown. As the amplitude of oscillation becomes damped over time, this apparent homeostasis no longer remains stable (Figs. 2(a), 3(a), and 4(a)). Low level static mechanical stimuli, 0.0≤*ξ*(*T*)≤1.0, makes the oscillatory network of anabolic and catabolic pathways further damped. As a consequence, the matrix molecules attain a much lower concentration at homeostasis. Results indicate the altered dynamics of anabolic and catabolic when the mechanical stimuli are increased beyond 1.0. Here, a higher mechanical stress threshold may dominate the anabolic pathway over catabolic actions (Fig. 5(c)), and higher transient values of matrix accumulation are expected.

Results presented here offer a unique perspective toward understanding the influence of mechanical stimulation on biomolecule homeostasis. Scientists have conducted extensive research to understand this process of mechanotransduction, especially with regard to cartilage biology. Unfortunately, very little is known regarding how mechanical loading modulates the dynamics of the biochemical activity through changes in the cellular kinetics. Multiaxial applied stresses at the mesoscale (tissue) level result in complex changes at the nanoscale (molecular) level. These influences include ECM and cell deformation, changes in hydrostatic pressure gradients and fluid flow, changes in ion concentration and fixed charged density, etc. [39]. The mechanical stimuli may be of different types such as structural deformation of intracellular biomolecules or kinetic changes in the interactions of different extracellular biomolecules, all resulting in cell-mediated biosynthesis. Chondrocyte mechanoreceptors such as mechanosensitive ion channels [40] and integrins [41] are likely to be involved in recognition of these physiochemical changes. Most of the research has focused on signaling pathways that transduce mechanical stimuli in cartilage [16,42–57]. These studies generally indicate that multiple signaling pathways are covariate and interactive within the mechanotransduction process. Recent advancements in cartilage tissue engineering research also advocate that there are multiple pathways by which chondrocytes sense and respond to mechanical stimuli [58]. Mechanical stimuli at the nanoscale can influence cells in different ways, such as by epigenetic regulatory mechanism that controls genetic transcription and translation processes. The level of influence may, in turn, affect cell-mediated matrix assembly and degradation. The study presented here has taken a systems biology approach to examine the multiscale influences on the biokinetics of mechanotransduction.

Here, we investigated the impact of static mechanical stress on two fundamental pathways of matrix homeostasis, namely, anabolic and catabolic pathways. Our model simulations show that there is a distinct decrease in matrix molecule accumulation during homeostasis if the static mechanical loading stimulus is very low. Similar results obtained by experimentalists conclude that static compression significantly inhibits the synthesis of proteoglycans and other protein molecules [59,60].

Although limited to the static loading case, our simulation predicts that a sustainable harmonic balance between the two important pathways may be the key to obtain a stable matrix homeostasis. The static loading model attempts to attain the matrix homeostasis but with very low accumulation compared to the absence of mechanical loading. Higher order static and dynamic mechanical stimulation may drive the modeled system toward attaining homeostasis at a much higher level of accumulation. Ongoing efforts will build on the presented mathematical framework and explore the complexities of these unique system dynamics.

Support was provided by the National Institutes of Health as a Research Infrastructure for Minority Institutions (RIMI) Exploratory Program grant (Grant No. P20MD003350) established the CSU Center for Allaying Health Disparities through Research and Education (CADRE) and a PSU Academic Research Enhancement Award (Grant No. R15EB007077). Support was also provided by the Collins Medical Trust and a Portland State University Faculty Enhancement Grant.

Asit K. Saha, Center for Allaying Health Disparities through Research and Education (CADRE), Department of Mathematics and Computer Science, Central State University, Wilberforce, OH 45384.

Sean S. Kohles, Reparative Bioengineering Laboratory, Department of Mechanical and Materials Engineering, Portland State University, Portland, OR 97201; Department of Surgery, Oregon Health and Science University, Portland, OR 97201.

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