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Exploring the Biochemical and Mechanical Effects of Intestinal Edema on Smooth Muscle Cell Contraction

Module by: Muhammad Shamim. E-mail the author

Summary: This study was done at the Department of Computational and Applied Mathematics at Rice University in the summer of 2011 under the supervision of Dr. Jennifer Young as part of the VIGRE Program.

Authors: Isadora Calderon, Drew Ferguson, Andria Remirez and Muhammad Shamim

VIGRE Mentor: Dr. Jennifer Young

Introduction

Intestinal edema is the accumulation of excess interstitial fluid in the intestinal wall tissue. It can occur as a consequence of resuscitative treatment given after traumatic injury [19]. After fluid resuscitation, the lymphatic system is unable to immediately remove the extra fluid from the interstitial spaces. The excess fluid is known to cause decreased smooth muscle cell (SMC) contractility, a condition referred to as ileus [20]. However, the connection between edema and decreased SMC contractility has not been clearly established. In this study, we seek to understand the connection by testing two hypotheses with mathematical models.

Due to increased interstitial fluid in edema, neurotransmitters at the neuromuscular junction must diffuse over greater synaptic cleft distances to reach receptors on the SMC membrane [16], [6]. The first hypothesis analyzes the effect of these increased distances across the synaptic cleft on the concentrations of the neurotransmitter acetylcholine (ACH). Increased interstitial fluid also causes uncoiling of collagen fibers in the extracellular matrix, mechanically straining the cell's contractile process. The second hypothesis analyzes the effect of this increased strain of the collagen fibers on SMC contraction.

In order to test these two hypotheses, a comprehensive computational model incorporating biochemical and mechanical interactions of the SMC was developed. Many existing biochemical models were incorporated into the comprehensive model, but few mechanical models of SMC contraction have been developed. Existing mechanical SMC models only model contraction without biochemical inputs. Unique to our comprehensive model was incorporation of ACH diffusion, actin-myosin powerstroking, the cell membrane and cytoskeleton, and the extracellular collagen fibers.

Biology

The intestines play an integral role in digestion. As chyme exits the stomach, it enters the small intestines where digestion is continued and nutrients are absorbed into the blood stream through microvilli [5]. These processes occur within a central hollow region of the intestinal tract known as the lumen. Surrounding the lumen is the intestinal wall, composed of various tissue layers. Among these is the muscularis externa, which is composed of two layers of smooth muscle tissue: the circular tissue layer and the longitudinal layer [5]. Digesting material is propelled through the intestines for eventual excretion by the coordinated contraction of these two smooth muscle layers in a unidirectional squeezing motion known as peristalsis [5]. It is this process that is interrupted by edema formation [11], [15].

The tissue that composes both of the muscular tissue layers consists of interconnected SMCs [13]. A SMC is roughly ellipsoid in shape, with a length of 100 to 300 microns and a width of 5 to 10 microns [17]. Upon stimulation by nerves present in the muscularis, a SMC will contract. This contraction can vary in magnitude, with the maximum extent of contraction estimated to be approximately 70% of the cell's resting length [17]. The cytoplasm of neighboring SMCs are often connected to one another via channels known as gap junctions, allowing for the spreading activation of chemical and electrical signals [10]. Consequently, stimulation of one cell by an agonist will result in the contraction of multiple SMCs due to the flow of chemicals from the originally activated cell to adjacent cells [10], [13].

Some of the mechanical components necessary for cellular contraction are actin thin filaments, myosin-II heavy filaments, intermediate filaments, and dense bodies and plaques [2]. Actin and myosin work in tandem in the mechanochemical transduction of force. Intermediate filaments and actin filaments function as cytoskeletal elements that provide structure to the cell as well as spread the force generated by actin and myosin around the entire cell membrane [2], [13]. Intermediate filaments also function as connecting filaments that transmit force between dense bodies and dense plaques [2], [13].

Actin and myosin directly generate the force of contraction. When intracellular calcium concentrations rise, a globular protein protruding from the myosin-II heavy filament known as the myosin head becomes activated [17], [2]. This head binds to an active site located along the length of the actin filament; multiple active sites exist along a single filament [2]. When ATP phosphorylates the myosin head, the structure undergoes a series of conformational changes that replicates a rowing motion, pulling the actin filament through the cytoplasm. This action is termed the powerstroke, which is estimated to move an actin filament 10 nm by exerting a force of 3-4 pN at the actin-myosin binding site [17], [13]. Cycles of this powerstroke motion result in a continuous pattern of latching and releasing along these binding sites that pulls the actin through the cell. Actin, in turn, is attached to dense bodies or dense plaques, to which the force of contraction is applied [2], [13]. Dense bodies (plaques) are nodal structures anchored in the cytoplasm (plasma membrane) and connected to contracting actin filaments [2]. When actin and myosin generate force, it is applied to these structures such that the SMC can undergo the conformational changes necessary for cellular contraction.

Actin and myosin powerstroking is regulated by a complex network of chemical reactions, as shown in Figure 1. Intracellular calcium (Ca2+Ca2+) is the primary chemical controlling the contraction-relaxation cycles [13], [17]. Four Ca2+Ca2+ ions bind to the cytoplasmic enzyme calmodulin, forming the calcium-calmodulin complex (Ca4CCa4C). Ca4CCa4C then binds to the enzyme myosin light-chain kinase (MKMK), activating its phosphorylation capabilities. This activated complex (Ca4CMKCa4CMK) phosphorylates the regulatory myosin light-chain (MLC20MLC20) located at the base of the myosin head. Once activated, this head can bind with ATP to initiate the cross-bridge cycling necessary for contraction [4], [17]. The myosin head's activation simultaneously is countered by the enzymatic activity of myosin light-chain phosphatase (MLML) [4]. The degree to which the cell contracts depends on the relative quantities of MLML and Ca4CMKCa4CMK in the cell. When the cell relaxes from decreasing intracellular [Ca2+][Ca2+], MLML deactivates MLC20MLC20 at a greater rate than its antagonist pair can trigger activation, disabling the myosin head's ability to produce the powerstroke.

Phosphorylated myosin (MpMp) binds to an actin thin filament (A), forming the actomyosin complex (AMpAMp). AMpAMp catalyzes the breakdown of ATP into ADP and PiPi (inorganic phosphate), which leads to a conformational change in the myosin head. Once PiPi is released from AMpAMp, the myosin head pulls the actin filament through the cytoplasm in an action termed the powerstroke. This stroking motion is the moment of mechanochemical force generation, and the process continues cyclically during contraction.

Hypothesis 1

SMC contraction can be initiated by agonist stimulation. The first hypothesis explored agonist stimulation in edematous conditions by ACH, as shown in figure 2, and its influence on the magnitude of cellular contraction. Neuron firing causes exocytosis of vesicles carrying ACH at the synaptic terminals. ACH then diffuses across the synaptic cleft to the plasma membrane of the target SMC. This distance is relatively small, with an average value of 12 - 20 nm [16]. ACH then binds to receptor proteins along the plasma membrane, which in turn leads to the formation of the secondary messenger Inositol 1,4,5-trisphosphate (IP3IP3) within the cell [9]. Activated IP3IP3 stimulates the release of calcium ions (Ca2+Ca2+) from the sarcoplasmic reticulum, and these calcium ions work as a positive feedback mechanism in the release of more Ca2+Ca2+ from the inner sarcoplasmic space, in a process known as calcium-induced calcium release (CICR) [9]. The increased calcium concentration affects the membrane potential of the cell along with potassium (K+K+) channels embedded in the plasma membrane [10]. Both the membrane potential and K+K+ channels regulate [Ca2+Ca2+] as well.

According to the first hypothesis, the chemical pathways are interrupted by intestinal edema. It was proposed that as the volume of fluid increases in the muscularis, the fluid penetrates into the interstitial spaces between SMCs and increases the distance between a target cell and the neuron that activates its contraction. As stated earlier, a typical synaptic cleft ranges from 12-20nm in length, but under edematous conditions this gap can reach anywhere from 30-60 nm [21]. Because of this larger synaptic distance, the volume through which the neurotransmitter ACH can diffuse also increases, and consequently the final concentration reaching the cell membrane receptors is lowered. The magnitude of the contractile response was expected to decrease due to decreased stimulation of secondary reaction networks. In testing hypothesis one, diffusion of ACH across the synaptic space was modeled using the two-dimensional heat equation under normal conditions and edematous conditions of varying magnitude. An explanation of this implementation is found in the discussion of the mathematical model (Section 2.1.1).

Hypothesis 2

Edema causes increased fluid in the interstitial spaces between SMCs, which overlap with the extracellular matrix. The extracellular matrix is composed of different proteins and collagen fibers. These fibers are attached to the cell membranes of the muscle cells and behave similar to coiled springs, able to stretch and compress easily, allowing the attached cells to contract and relax [12].

However due to excess fluid in the interstitial spaces in edema, the collagen fibers become stretched and uncoiled, increasing the elastic modulus for the collagen fibers. This increased elastic modulus was expected to decrease the magnitude of contraction in SMCs. In testing hypothesis two, additional mechanical tension was placed in the collagen fibers in the extracellular matrix to analyze changes in the magnitude of cellular contraction.

Mathematical Model

Biochemical Model

ACH Simulation

ACH flow was simulated by using forward finite difference for time and central finite difference for space to find an explicit solution to two-dimensional heat flow. The diffusion coefficient of ACH was calculated from [18].

[ A C H ] t = α [ A C H ] x x + [ A C H ] y y α = 0 . 4 n m 2 n s - 1 [ A C H ] t = α [ A C H ] x x + [ A C H ] y y α = 0 . 4 n m 2 n s - 1
(1)

It was found experimentally that using a time step of 0.25 ns for explicit solution gave similar results as using a Gaussian lowpass filter matrix to represent a rotational symmetric point surface distribution. Since the Gaussian filter matrix, B, is built into MATLAB, it is computed faster, and thus was used for the simulation. In equation 2, B is the 3 by 3 matrix from a Gaussian lowpass filter matrix with σ=0.4σ=0.4. In MATLAB, B=fspecial('gaussian',3,0.4)B=fspecial('gaussian',3,0.4). Time is scaled to increments of 0.25 ns.

[ A C H ] x , y t + 1 = ∑ i = - 1 1 ∑ j = - 1 1 β i , j [ A C H ] x + i , y + j t β i , j = B ( 2 + i , 2 + j ) [ A C H ] x , y t + 1 = ∑ i = - 1 1 ∑ j = - 1 1 β i , j [ A C H ] x + i , y + j t β i , j = B ( 2 + i , 2 + j )
(2)

The synaptic cleft was represented by a two-dimensional grid of length 100 nm. The width was varied from 20 to 60 nm to represent the changing distance in edematous conditions. Each cell (x, y) on the grid was 1 nm ×× 1 nm, and contained the concentration for that cell at a given time (t), represented in equation 2 as [ACH]x,yt[ACH]x,yt. The initial concentration at each cell was 0 μMμM, except at the neuron terminal, where the ACH was released by exocytosis at concentrations of 1000 μMμM in the row of cells from [ACH]45,2[ACH]45,2 to [ACH]54,2[ACH]54,2. Homogeneous Dirichlet boundary conditions were used at all boundaries except at the release site and receptor site, which were made impenetrable. These impenetrable boundaries were the row of cells from [ACH]36,1[ACH]36,1 to [ACH]63,1[ACH]63,1 (neuron release site) and [ACH]36,ζ[ACH]36,ζ to [ACH]63,ζ[ACH]63,ζ (membrane receptor site), where ζζ was the synaptic cleft width.

Calcium Control

ACH receptors influence IP3 levels, which affect cell potential, potassium channel probability, SR calcium, and intracellular calcium. The IP3 model was adapted from [9] and the remaining differential equations were adapted from [10].

d [ I P 3 ] d t = [ A C H ] - ϵ [ I P 3 ] - V M 4 [ I P 3 ] u [ I P 3 ] u + K 4 u + P M V ( 1 - [ E ] r 2 ) K V r 2 + [ E ] r 2 d [ C a S R 2 + ] d t = B [ C a 2 + ] 2 [ C a 2 + ] 2 + C b 2 - C [ C a S R 2 + ] 2 [ C a 2 + ] 4 ( [ C a S R 2 + ] 2 + s c 2 ) ( [ C a 2 + ] 4 + c c 4 ) - L [ C a S R 2 + ] d [ W ] d t = λ ( [ C a 2 + ] + c W ) 2 ( [ C a 2 + ] + c W ) 2 + β e - ( [ E ] - v C a 3 R K ) - [ W ] d [ E ] d t = γ ( - F N a K - G C l ( [ E ] - v C l ) - 2 G C a ( [ E ] - v C a 1 ) 1 + e - [ E ] - v C a 2 R C a - G N C X [ C a 2 + ] ( [ E ] - v N C X ) [ C a 2 + ] + c N C X - G K [ W ] ( [ E ] - v K ) ) d [ C a 2 + ] d t = F [ I P 3 ] 2 [ I P 3 ] 2 + K r 2 - G C a ( [ E ] - v C a 1 1 + e - ( [ E ] - v C a 2 R C a ) + G N C X [ C a 2 + ] ( [ E ] - v N C X ) [ C a 2 + ] + c N C X - B [ C a 2 + ] 2 [ C a 2 + ] 2 + C b 2 + C [ C a S R 2 + ] 2 [ C a 2 + ] 4 ( [ C a S R 2 + ] 2 + s c 2 ) ( [ C a 2 + ] 4 + c c 4 ) - D [ C a 2 + ] ( 1 + ( [ E ] - v d ) R d + L [ C a S R 2 + ] d [ I P 3 ] d t = [ A C H ] - ϵ [ I P 3 ] - V M 4 [ I P 3 ] u [ I P 3 ] u + K 4 u + P M V ( 1 - [ E ] r 2 ) K V r 2 + [ E ] r 2 d [ C a S R 2 + ] d t = B [ C a 2 + ] 2 [ C a 2 + ] 2 + C b 2 - C [ C a S R 2 + ] 2 [ C a 2 + ] 4 ( [ C a S R 2 + ] 2 + s c 2 ) ( [ C a 2 + ] 4 + c c 4 ) - L [ C a S R 2 + ] d [ W ] d t = λ ( [ C a 2 + ] + c W ) 2 ( [ C a 2 + ] + c W ) 2 + β e - ( [ E ] - v C a 3 R K ) - [ W ] d [ E ] d t = γ ( - F N a K - G C l ( [ E ] - v C l ) - 2 G C a ( [ E ] - v C a 1 ) 1 + e - [ E ] - v C a 2 R C a - G N C X [ C a 2 + ] ( [ E ] - v N C X ) [ C a 2 + ] + c N C X - G K [ W ] ( [ E ] - v K ) ) d [ C a 2 + ] d t = F [ I P 3 ] 2 [ I P 3 ] 2 + K r 2 - G C a ( [ E ] - v C a 1 1 + e - ( [ E ] - v C a 2 R C a ) + G N C X [ C a 2 + ] ( [ E ] - v N C X ) [ C a 2 + ] + c N C X - B [ C a 2 + ] 2 [ C a 2 + ] 2 + C b 2 + C [ C a S R 2 + ] 2 [ C a 2 + ] 4 ( [ C a S R 2 + ] 2 + s c 2 ) ( [ C a 2 + ] 4 + c c 4 ) - D [ C a 2 + ] ( 1 + ( [ E ] - v d ) R d + L [ C a S R 2 + ]
(3)
Table 1
[ A C H ] 0 = [ A C H ] 0 = 0 . 001 μ M 0 . 001 μ M Initial ACH concentration  
[ I P 3 ] 0 = [ I P 3 ] 0 = 0 . 49 μ M 0 . 49 μ M Initial IP3 concentration  
[ C a S R 2 + ] 0 = [ C a S R 2 + ] 0 = 1 . 1 μ M 1 . 1 μ M Initial sarcoplasmic calcium concentration  
[ W ] 0 = [ W ] 0 = 0 . 02 0 . 02 Initial potassium channel probability concentration  
[ E ] 0 = [ E ] 0 = - 42 m V - 42 m V Initial cell potential  
[ C a 2 + ] 0 = [ C a 2 + ] 0 = 0 . 17 μ M 0 . 17 μ M Initial intracellular calcium concentration  
β = β = 0 . 13 μ M 2 0 . 13 μ M 2 translation factor [10]
γ = γ = 197 m V μ M - 1 197 m V μ M - 1 scaling factor [10]
ϵ = ϵ = 0 . 015 s - 1 0 . 015 s - 1 rate constant for linear IP3 [9]
λ = λ = 45 channel constant [10]
B = B = 2 . 025 μ M s - 1 2 . 025 μ M s - 1 SR uptake rate constant [10]
C = C = 55 μ M s - 1 55 μ M s - 1 CICR rate constant [10]
D = D = 0 . 24 s - 1 0 . 24 s - 1 Ca extrusion by ATPase constant [10]
F = F = 0 . 23 μ M s - 1 0 . 23 μ M s - 1 maximal influx rate [10]
L = L = 0 . 025 s - 1 0 . 025 s - 1 leak from SR rate constant [10]
C b = C b = 1 μ M 1 μ M half point SR ATPase activation [10]
c c = c c = 0 . 9 μ M 0 . 9 μ M half point CICR activation [10]
c N C X = c N C X = 0 . 5 μ M 0 . 5 μ M half point Na Ca exchange activation [10]
c W = c W = 0 . 0 μ M 0 . 0 μ M translation factor [10]
F N a K = F N a K = 0 . 0432 μ M s - 1 0 . 0432 μ M s - 1 net whole cell flux [10]
G C a = G C a = 0 . 00129 μ M m V - 1 s - 1 0 . 00129 μ M m V - 1 s - 1 whole cell conductance for VOCCs [10]
G C l = G C l = 0 . 00134 μ M m V - 1 s - 1 0 . 00134 μ M m V - 1 s - 1 whole cell conductance Cl [10]
G K = G K = 0 . 00446 μ M m V - 1 s - 1 0 . 00446 μ M m V - 1 s - 1 whole cell conductance K [10]
G N C X = G N C X = 0 . 00316 μ M m V - 1 s - 1 0 . 00316 μ M m V - 1 s - 1 whole cell conductance for Na Ca exchange [10]
K 4 = K 4 = 0 . 5 μ M 0 . 5 μ M half saturation constant IP3 degradation [9]
K r = K r = 1 μ M 1 μ M half saturation constant Ca entry [10]
K V = K V = - 58 m V - 58 m V half saturation constant IP3 voltage synthesis [9]
P M V = P M V = 0 . 01333 μ M s - 1 0 . 01333 μ M s - 1 max rate voltage IP3 synthesis [9]
R 2 = R 2 = 8 hill coefficient [9]
R C a = R C a = 8 . 5 m V 8 . 5 m V maximum slope of VOCC activation [10]
R d = R d = 250 . 0 m V 250 . 0 m V slope of voltage dependence [10]
R K = R K = 12 . 0 m V 12 . 0 m V maximum slope Ca activation [10]
s c = s c = 2 μ M 2 μ M half point CICR efflux [10]
u = u = 4 hill coefficient [9]
v C a 1 = v C a 1 = 100 . 0 m V 100 . 0 m V reversal potential VOCCs [10]
v C a 2 = v C a 2 = - 24 m V - 24 m V half point VOCC activation [10]
v C a 3 = v C a 3 = - 27 m V - 27 m V half point Ca channel activation [10]
v C l = v C l = - 25 m V - 25 m V reversal potential Cl [10]
v d = v d = - 100 . 0 m V - 100 . 0 m V intercept voltage dependence [10]
v K = v K = - 104 . 0 m V - 104 . 0 m V reversal potential K [10]
V M 4 = V M 4 = 0 . 0333 μ M s - 1 0 . 0333 μ M s - 1 max nonlinear IP degradation [9]
v N C X = v N C X = - 40 . 0 m V - 40 . 0 m V reversal potential Na Ca exchange [10]

Myosin Activation

The activation of the MKMK complex includes 4 Ca2+Ca2+ ions bonded to calmodulin and myosin light chain kinase. Side reactions include the disassociation of the kinase from the complex and interactions with binding proteins. These interactions were modeled using mass action kinetics summarized in 9 reactions from [4].

d [ C ] d t = r 1 [ C a 2 C ] + r 3 [ C M K ] + r 8 [ C B P ] - f 1 [ C a 2 + ] 2 [ C ] - f 3 [ C ] [ M K ] - f 8 [ C ] [ B P ] d [ M K ] d t = r 3 [ C M K ] + r 4 [ C a 2 C M K ] + r 5 [ C a 4 C M K ] - f 3 [ C ] [ M K ] - f 4 [ M K ] [ C a 2 C ] - f 5 [ M K ] [ C a 4 C ] d [ C a 2 C ] d t = f 1 [ C a 2 + ] 2 [ C ] + r 2 [ C a 4 C ] + r 4 [ C a 2 C M K ] + f 9 [ C a 2 + ] 2 [ C B P ] - r 1 [ C a 2 C ] - f 2 [ C a 2 + ] 2 [ C a 2 C ] - f 4 [ M K ] [ C a 2 C ] - r 9 [ C a 2 C ] [ B P ] d [ C a 4 C ] d t = f 2 [ C a 2 + ] 2 [ C a 2 C ] + r 5 [ C a 4 C M K ] - r 2 [ C a 4 C ] - f 5 [ M K ] [ C a 4 C ] d [ C M K ] d t = f 3 [ C ] [ M K ] + r 6 [ C a 2 C M K ] - r 3 [ C M K ] - f 6 [ C a 2 + ] 2 [ C M K ] d [ C a 2 C M K ] d t = f 4 [ M K ] [ C a 2 C ] + f 6 [ C a 2 + ] 2 [ C M K ] + r 7 [ C a 4 C M K ] - r 4 [ C a 2 C M K ] - r 6 [ C a 2 C M K ] - f 7 [ C a 2 + ] 2 [ C a 2 C M K ] d [ C a 4 C M K ] d t = f 5 [ M K ] [ C a 4 C ] + f 7 [ C a 2 + ] 2 [ C a 2 C M K ] - r 5 [ C a 4 C M K ] - r 7 [ C a 4 C M K ] d [ B P ] d t = r 8 [ C B P ] + f 9 [ C a 2 + ] 2 [ C B P ] - f 8 [ C ] [ B P ] - r 9 [ C a 2 C ] [ B P ] d [ C B P ] d t = f 8 [ C ] [ B P ] + r 9 [ C a 2 C ] [ B P ] - r 8 [ C B P ] - f 9 [ C a 2 + ] 2 [ C B P ] d [ C ] d t = r 1 [ C a 2 C ] + r 3 [ C M K ] + r 8 [ C B P ] - f 1 [ C a 2 + ] 2 [ C ] - f 3 [ C ] [ M K ] - f 8 [ C ] [ B P ] d [ M K ] d t = r 3 [ C M K ] + r 4 [ C a 2 C M K ] + r 5 [ C a 4 C M K ] - f 3 [ C ] [ M K ] - f 4 [ M K ] [ C a 2 C ] - f 5 [ M K ] [ C a 4 C ] d [ C a 2 C ] d t = f 1 [ C a 2 + ] 2 [ C ] + r 2 [ C a 4 C ] + r 4 [ C a 2 C M K ] + f 9 [ C a 2 + ] 2 [ C B P ] - r 1 [ C a 2 C ] - f 2 [ C a 2 + ] 2 [ C a 2 C ] - f 4 [ M K ] [ C a 2 C ] - r 9 [ C a 2 C ] [ B P ] d [ C a 4 C ] d t = f 2 [ C a 2 + ] 2 [ C a 2 C ] + r 5 [ C a 4 C M K ] - r 2 [ C a 4 C ] - f 5 [ M K ] [ C a 4 C ] d [ C M K ] d t = f 3 [ C ] [ M K ] + r 6 [ C a 2 C M K ] - r 3 [ C M K ] - f 6 [ C a 2 + ] 2 [ C M K ] d [ C a 2 C M K ] d t = f 4 [ M K ] [ C a 2 C ] + f 6 [ C a 2 + ] 2 [ C M K ] + r 7 [ C a 4 C M K ] - r 4 [ C a 2 C M K ] - r 6 [ C a 2 C M K ] - f 7 [ C a 2 + ] 2 [ C a 2 C M K ] d [ C a 4 C M K ] d t = f 5 [ M K ] [ C a 4 C ] + f 7 [ C a 2 + ] 2 [ C a 2 C M K ] - r 5 [ C a 4 C M K ] - r 7 [ C a 4 C M K ] d [ B P ] d t = r 8 [ C B P ] + f 9 [ C a 2 + ] 2 [ C B P ] - f 8 [ C ] [ B P ] - r 9 [ C a 2 C ] [ B P ] d [ C B P ] d t = f 8 [ C ] [ B P ] + r 9 [ C a 2 C ] [ B P ] - r 8 [ C B P ] - f 9 [ C a 2 + ] 2 [ C B P ]
(4)
Table 2
[ C ] 0 = [ C ] 0 = 0 . 9285 μ M 0 . 9285 μ M   Initial Calmodulin (C) concentration [4]
[ M K ] 0 = [ M K ] 0 = 9 . 6506 μ M 9 . 6506 μ M   Initial Myosin LC Kinase (MKMK) concentration [4]
[ C a 2 C ] 0 = [ C a 2 C ] 0 = 0 . 0015 μ M 0 . 0015 μ M   Initial Ca2CCa2C complex concentration [4]
[ C a 4 C ] 0 = [ C a 4 C ] 0 = 0 . 00 μ M 0 . 00 μ M   Initial Ca4CCa4C complex concentration [4]
[ C M K ] 0 = [ C M K ] 0 = 0 . 3332 μ M 0 . 3332 μ M   Initial CMKCMK complex concentration [4]
[ C a 2 C M K ] 0 = [ C a 2 C M K ] 0 = 0 . 2713 μ M 0 . 2713 μ M   Initial Ca2CMKCa2CMK complex concentration [4]
[ C a 4 C M K ] 0 = [ C a 4 C M K ] 0 = 0 . 013 μ M 0 . 013 μ M   Initial Ca4CMKCa4CMK activated complex concentration [4]
[ B P ] 0 = [ B P ] 0 = 15 . 1793 μ M 15 . 1793 μ M   Initial Binding Protein (BPBP) concentration [4]
[ C B P ] 0 = [ C B P ] 0 = 2 . 8207 μ M 2 . 8207 μ M   Initial C-BPC-BP complex concentration [4]
[ f 1 r 1 ] = [ f 1 r 1 ] = [ 12 μ M - 1 [ 12 μ M - 1 12 ] s - 1 12 ] s - 1 Forward and reverse rates for reaction 1 [4]
[ f 2 r 2 ] = [ f 2 r 2 ] = [ 480 μ M - 1 [ 480 μ M - 1 1200 ] s - 1 1200 ] s - 1 Forward and reverse rates for reaction 2 [4]
[ f 3 r 3 ] = [ f 3 r 3 ] = [ 5 μ M - 1 [ 5 μ M - 1 135 ] s - 1 135 ] s - 1 Forward and reverse rates for reaction 3 [4]
[ f 4 r 4 ] = [ f 4 r 4 ] = [ 840 μ M - 1 [ 840 μ M - 1 45 . 4 ] s - 1 45 . 4 ] s - 1 Forward and reverse rates for reaction 4 [4]
[ f 5 r 5 ] = [ f 5 r 5 ] = [ 28 μ M - 1 [ 28 μ M - 1 0 . 0308 ] s - 1 0 . 0308 ] s - 1 Forward and reverse rates for reaction 5 [4]
[ f 6 r 6 ] = [ f 6 r 6 ] = [ 120 μ M - 1 [ 120 μ M - 1 4 ] s - 1 4 ] s - 1 Forward and reverse rates for reaction 6 [4]
[ f 7 r 7 ] = [ f 7 r 7 ] = [ 7 . 5 μ M - 1 [ 7 . 5 μ M - 1 3 . 75 ] s - 1 3 . 75 ] s - 1 Forward and reverse rates for reaction 7 [4]
[ f 8 r 8 ] = [ f 8 r 8 ] = [ 5 μ M - 1 [ 5 μ M - 1 25 ] s - 1 25 ] s - 1 Forward and reverse rates for reaction 8 [4]
[ f 9 r 9 ] = [ f 9 r 9 ] = [ 7 . 6 μ M - 1 [ 7 . 6 μ M - 1 22 . 8 ] s - 1 22 . 8 ] s - 1 Forward and reverse rates for reaction 9 [4]

Force Generation

The interactions between myosin, actin, and the activated MKMK complex were modeled using Henri-Michaelis-Menten Enzyme Kinetics from [4].

d [ M ] d t = - k 1 [ C a 4 C M K ] [ M ] k 2 + [ M ] + k 5 [ M L ] [ M p ] k 6 + [ M p ] + k 7 [ A M ] d [ M p ] d t = k 1 [ C a 4 C M K ] [ M ] k 2 + [ M ] - k 5 [ M L ] [ M p ] k 6 + [ M p ] - k 3 [ M p ] + k 4 [ A M p ] d [ A M p ] d t = k 3 [ M p ] - k 4 [ A M p ] + k 1 [ C a 4 C M K ] [ A M ] k 2 + [ A M ] - k 5 [ M L ] [ A M p ] k 6 + [ A M p ] d [ A M ] d t = - k 1 [ C a 4 C M K ] [ A M ] k 2 + [ A M ] + k 5 [ M L ] [ A M p ] k 6 + [ A M p ] - k 7 [ A M ] F ( t ) = F m a x [ A M ( t ) ] + [ A M p ( t ) ] [ M T ] d [ M ] d t = - k 1 [ C a 4 C M K ] [ M ] k 2 + [ M ] + k 5 [ M L ] [ M p ] k 6 + [ M p ] + k 7 [ A M ] d [ M p ] d t = k 1 [ C a 4 C M K ] [ M ] k 2 + [ M ] - k 5 [ M L ] [ M p ] k 6 + [ M p ] - k 3 [ M p ] + k 4 [ A M p ] d [ A M p ] d t = k 3 [ M p ] - k 4 [ A M p ] + k 1 [ C a 4 C M K ] [ A M ] k 2 + [ A M ] - k 5 [ M L ] [ A M p ] k 6 + [ A M p ] d [ A M ] d t = - k 1 [ C a 4 C M K ] [ A M ] k 2 + [ A M ] + k 5 [ M L ] [ A M p ] k 6 + [ A M p ] - k 7 [ A M ] F ( t ) = F m a x [ A M ( t ) ] + [ A M p ( t ) ] [ M T ]
(5)
Table 3
[ M ] 0 = [ M ] 0 = 23 . 9558 μ M 23 . 9558 μ M Initial myosin concentration [4]
[ M p ] 0 = [ M p ] 0 = 0 . 0144 μ M 0 . 0144 μ M Initial phosphorylated myosin concentration [4]
[ A M p ] 0 = [ A M p ] 0 = 0 . 0166 μ M 0 . 0166 μ M Initial cross-bride concentration [4]
[ A M ] 0 = [ A M ] 0 = 0 . 0132 μ M 0 . 0132 μ M Initial latch-bridge concentration [4]
[ M L ] = [ M L ] = 7 . 5 μ M 7 . 5 μ M Myosin light chain phosphatase concentration [4]
[ M T ] = [ M T ] = 24 μ M 24 μ M Total myosin concentration [4]
F ( t ) = F ( t ) =   Force generated in mNmN [4]
F m a x = F m a x = 70 m N 70 m N Maximum force cell can generate [4]
k 1 = k 1 = 27 s - 1 27 s - 1 Rate for reaction 10 [4]
k 2 = k 2 = 10 μ M 10 μ M Rate for reaction 11 [4]
k 3 = k 3 = 15 s - 1 15 s - 1 Forward rate for reaction 12 [4]
k 4 = k 4 = 5 s - 1 5 s - 1 Reverse rate for reaction 12 [4]
k 5 = k 5 = 16 s - 1 16 s - 1 Rate for reaction 13 [4]
k 6 = k 6 = 15 μ M 15 μ M Rate for reaction 14 [4]
k 7 = k 7 = 10 s - 1 10 s - 1 Rate for reaction 15 [4]

Mechanical Model

A variety of models which represent SMCs and other types of cells as mass-spring systems have been developed [7], [1], [8], [21], [10], [14]. The most comparable of these models used a single contractile element and two passive elements: one to represent the elastic actin and myosin fibers and the other to represent the adjacent muscle cells and surroundings [7]. We present a novel mechanical model of the SMC which incorporates the previously described biochemical interactions to produce a comprehensive model of SMC contraction.

The SMC is modeled as a dynamic two-dimensional mass-spring system. The membrane and the various filaments of the cell are discretized into a series of springs with mass distributed amongst nodes located at the intersections of these springs. The spring stiffnesses and node masses vary depending on the material they represent.

The model includes the powerstroking process with myosin heads extending from nodes along the myosin filament and connecting to nodes along the actin filaments. The contractile force generated by the biochemical model is applied at these actin binding sites, in a direction parallel to the axis of the actin filament. After the duration of a typical powerstroke has passed, each myosin head detaches from its corresponding binding site on actin, and reattaches to the actin at the next node on the filament. The location at which the contractile force is applied changes according to which actin nodes are attached to the myosin heads at the current time in the simulation. Figure 4 depicts the powerstroking process as used in the model.

The filaments which participate in the powerstroking process are considered active filaments, as they are the components of the cell which produce the cellular contraction. The model also includes several types of passive elements, including the cell membrane, the cytoskeletal network, and the extracellular matrix of collagen fibers. Unlike the active filaments, these passive elements do not produce a contractile force based on a series of biochemical reactions, but rather experience forces only as a result of drag, spring deformation, and the cell's own hydrostatic pressure.

Thus the model gives rise to the following equations which describe the motion of the ithith node iin the system:

∂ x i ∂ t = u i ∂ y i ∂ t = v i m i ∂ u i ∂ t = ∑ j = 1 N k i j ( L i j - L 0 i j ) x i - x j L i j + P A n x + F m y o s i n , x - β i u i m i ∂ v i ∂ t = ∑ j = 1 N k i j ( L i j - L 0 i j ) y i - y j L i j + P A n y + F m y o s i n , y - β i v i ∂ x i ∂ t = u i ∂ y i ∂ t = v i m i ∂ u i ∂ t = ∑ j = 1 N k i j ( L i j - L 0 i j ) x i - x j L i j + P A n x + F m y o s i n , x - β i u i m i ∂ v i ∂ t = ∑ j = 1 N k i j ( L i j - L 0 i j ) y i - y j L i j + P A n y + F m y o s i n , y - β i v i
(6)

Equations 22 and 23 represent the velocity of the node in the x- and y-directions, respectively; xixi is the x-coordinate of the position of the node, yiyi is the y-coordinate of the position of the node, uiui is the velocity in the x-direction and vivi is the velocity in the y-direction. Equations 24 and 25 are the equations of Newton's second law, again in the x- and y-directions. In these equations, mimi is the mass of node ii, NN is the number of other nodes which are attached to node ii by a spring, kijkij is the spring constant of the spring connecting nodes ii and jj, LijLij is the current length of that spring, and L0ijL0ij is the equilibrium length of that spring. PP is the hydrostatic pressure of the cell, AA is the cell's surface area, and nxnx and nyny are the x- and y-components of a unit vector normal to the cell membrane at node ii. Fmyosin,xFmyosin,x and Fmyosin,yFmyosin,y are the x- and y-components of the force on node ii generated by the biochemical reactions, and ββ is the drag coefficient for node ii. The right-hand sides of these last two equations represent the sum of the forces on the node.

Taking equation 24, the equation for motion in the x-direction, as an example, the first term on the right side of the equation represents the sum of the x-components of the spring forces of any filaments attached to that node as determined by Hooke's Law. The second term represents the x-component of the force acting on the node due to the higher pressure inside the cell. This force is only nonzero in the case of membrane nodes, and it is directed along a line normal to the cell membrane in an outward direction. The magnitude of the pressure force is inversely proportional to the area enclosed by the membrane filaments at the given time. The third term is the x-component of the powerstroking force (as previously described), which applies only to nodes along actin filaments which are connected to myosin heads at the current time. The final term in the equation represents the x-component of the force resulting from drag as the node moves through the cytoplasm. The drag coefficient, ββ, is approximated using slender-body theory [3]. Each of these terms has a y-directional analog which can be found in equation 25.

Results

The contractile behavior of the cell under normal conditions is depicted in figures 4 and 5. The figures show that contraction occurs after calcium levels have increased. The initial width of the relaxed cell under normal conditions was 200 microns, and the width of the cell at maximum contraction was 144.6 microns.

In testing hypothesis 1, shown in figure 6, increased synaptic cleft distance decreased ACH concentration at the cell membrane receptor site. In testing hypothesis 2, shown in figure 7, the cell's length at maximum contraction increased with an increase in interstitial area due to edema.

Discussion

The goal of this study was to explore a possible mechanical relationship between edema and ileus in intestinal SMCs. We have presented two hypotheses that may help explain the link between edema and ileus and developed a biochemical and mechanical model which analyzes the respective hypotheses. The model successfully replicates cellular contraction in non-edematous conditions. It also replicates the expected results of the two hypotheses, demonstrating decreased ACH levels at larger synaptic cleft widths and decreased contraction with stretched collagen fibers.

The next step in the project is to quantify the decrease in contraction from ACH decrease and collagen uncoiling. Further considerations include correcting ACH levels using ACHesterase or flow in three-dimensional space. These final results must then be compared to the decreased muscle activity in intestinal edema, to test if our model can satisfactorily explain the level of decreased activity.

References

  1. Bottino, D.C. and Fauci, L.J. (1998). A computational model of ameboid deformation and locomotion. European biophysics journal, 27(5), 532–539.
  2. Bray, D. (2001). Cell movements: from molecules to motility. Routledge.
  3. Cox, RG. (1970). The motion of long slender bodies in a viscous fluid Part 1. General theory. Journal of Fluid mechanics, 44(04), 791–810.
  4. Gajendiran, V. and Buist, M.L. (2011). A quantitative description of active force generation in gastrointestinal smooth muscle. International Journal for Numerical Methods in Biomedical Engineering.
  5. Greger, R. and Windhorst, U. (1996). Comprehensive human physiology: from cellular mechanisms to integration. Springer.
  6. Gunst, S.J. and Zhang, W. (2008). Actin cytoskeletal dynamics in smooth muscle: a new paradigm for the regulation of smooth muscle contraction. American Journal of Physiology-Cell Physiology, 295(3), C576.
  7. Hill, AV. (1970). First and last experiments in muscle mechanics. London and New York.
  8. Head, D.A. and Levine, A.J. and MacKintosh, FC. (2003). Deformation of cross-linked semiflexible polymer networks. Physical review letters, 91(10), 108102.
  9. Imtiaz, M.S. and Smith, D.W. and Van Helden, D.F. (2002). A theoretical model of slow wave regulation using voltage-dependent synthesis of inositol 1, 4, 5-trisphosphate. Biophysical journal, 83(4), 1877–1890.
  10. Koenigsberger, M. and Sauser, R. and Lamboley, M. and Bény, J.L. and Meister, J.J. (2004). Ca2+ dynamics in a population of smooth muscle cells: modeling the recruitment and synchronization. Biophysical journal, 87(1), 92–104.
  11. Moore-Olufemi, SD and Xue, H. and Allen, SJ and Moore, FA and Stewart, RH and Laine, GA and Cox, CS. (2005). Inhibition of Intestinal Transit by Resuscitation-Induced Gut Edema is Reversed by L-NIL1, 21. Journal of Surgical Research, 129(1), 1–5.
  12. MacKenna, DA and Vaplon, SM and McCULLOCH, A.D. (1997). Microstructural model of perimysial collagen fibers for resting myocardial mechanics during ventricular filling. American Journal of Physiology-Heart and Circulatory Physiology, 273(3), H1576.
  13. Pollard, TD and Earnshaw, WC and Lippincott-Schwartz, J. Cell Biology. 2nd.
  14. Robertson-Dunn, B. and Linkens, DA. (1974). A mathematical model of the slow-wave electrical activity of the human small intestine. Medical and Biological Engineering and Computing, 12(6), 750–758.
  15. Radhakrishnan, R.S. and Xue, H. and Weisbrodt, N. and Moore, F.A. and Allen, S.J. and Laine, G.A. and Cox Jr, C.S. (2005). Resuscitation-induced intestinal edema decreases the stiffness and residual stress of the intestine. Shock, 24(2), 165.
  16. Savtchenko, L.P. and Rusakov, D.A. (2007). The optimal height of the synaptic cleft. Proceedings of the National Academy of Sciences, 104(6), 1823.
  17. Schroeder, F. and Wood, WG and Kier, AB. (2001). In Cell Physiology Sourcebook: A Molecular Approach (Sperelakis, N., ed.).
  18. Tai, K. and Bond, S.D. and MacMillan, H.R. and Baker, N.A. and Holst, M.J. and McCammon, J.A. (2003). Finite element simulations of acetylcholine diffusion in neuromuscular junctions. Biophysical journal, 84(4), 2234–2241.
  19. Uray, K.S. and Laine, G.A. and Xue, H. and Allen, S.J. and Cox Jr, C.S. (2006). Intestinal edema decreases intestinal contractile activity via decreased myosin light chain phosphorylation. Critical care medicine, 34(10), 2630.
  20. Uray, K.S. and Laine, G.A. and Xue, H. and Allen, S.J. and Cox Jr, C.S. (2007). Edema-induced intestinal dysfunction is mediated by STAT3 activation. Shock, 28(2), 239.
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