can predict residual force enhancement on the descending limb of the length– tension curve in muscles during eccentric contraction. A kinematic model of titin. Within muscle there is also a relationship between the length of the muscle and . Next, we need to take into consideration the effect of the length-tension curve. The resting length of our muscles produces maximum tension. Length-Tension Curve. Microscopic Anatomy. We need to look at the minute.
Because you can see that our titin, which is in green, is really not allowing any space. Or there is no space, really. And so, these ends, remember these are our z-discs right here.
This is Z and this is Z over here. Our z-discs are right up against our myosin. In fact, there's almost no space in here. This is all crowded on both sides. There's no space for the myosins to actually pull the z-disc any closer. So because there's no space for them to work, they really can't work. And really, if you give them ATP and say, go to work.
They're going to turn around and say, well, we've got no work to do, because the z-disc is already here. So in terms of force of contraction for this scenario one, I would say, you're going to get almost no contraction.
So when the length is very low, so let's say this is low. Maybe low is not a good word for length. Let's say this is, I'll use the word short. The sarcomere is short. And here the sarcomere is long. So when it's short, meaning this distance is actually very short, then we would say the amount of tension is going to be actually zero. Because you really can't get any tension started unless you have a little bit of space between the z-disc and the myosin. So now in scenario two, let's say this is scenario two.
And this is my one circle over here.
Length tension relationship | S&C Research
In scenario two, what happens? Well, here you have a little bit more space, right? So let's draw that. Let's draw a little bit more space. Let's say you've got something like that. And I'm going to draw the other actin on this side, kind of equally long, of course. I didn't draw that correctly. Because if it's sliding out, you're going to have an extra bit of actin, right? And it comes up and over like that. So this is kind of what the actin would look like.
And, of course, I want to make sure I draw my titin. Titin is kind of helpful, because it helps demonstrate that there's now a little bit of space there where there wasn't any before. And so now there is some space between the z-disc and this myosin right here. So there is some space between these myosins and the z-discs. In fact, I can draw arrows all the way around.
And so there is a little bit of work to be done. But I still wouldn't say that it's maximal force. Because look, you still have some overlap issues. Remember, these myosins, right here, they're not able to work.
Biomechanical Hill muscle model
And neither are these, because of this blockage that's happening here. Because of the fact that, of course, actin has a certain polarity. So they're getting blocked. They can't do their work. And so even though you get some force of contraction, it wouldn't be maximal. So I'll put something like this. This will be our second spot. This will be number two. Now in number three, things are going to get much better. So you'll see very quickly now you have a much more spread out situation.
Where now these are actually-- these actins are really not going to be in the way of each other. You can see they're not bumping into each other, they're not in the way of each other at all.
And so all of the myosins can get to work. So the z-discs are now out here. My overall sarcomere, of course, as I said, was from z-disc to z-disc. So my sarcomere is getting longer. And you can also see that because now there's more titin, right? And there isn't actually more titin.Frank Starling Mechanism
I shouldn't use that phrase. But the titin is stretched out. So here, more work is going to get done. And now my force, I would say, is maximal. So I've got lots, and lots of force finally. And so it would be something like this. And so based on my curve, I've also demonstrated another point, which is that, the first issue, getting us from point one to point two, really helped a lot. I mean, that was the big, big deal. Because you needed some space here.
Again, this space really was necessary to do work at all. And now that we've gotten rid of the overlap issue, now that we've gotten these last few myosins working, we have even more gain. But the gain was really-- the biggest advantage was in that first step. Now as we go on, let's go to step four. So this is step four now. It is a purely mechanical muscle model built from the systems engineering perspective. The early pioneers in the field of muscle mechanics knew next to nothing about the internal structure and functioning of the muscle, so they had to treat it as a black box and attempted to map the input-output characteristics of muscle in an effort to formulate a mathematical model that could predict its overall behavior.
The second type of muscle model was formulated once the sliding filament theory was proposed by Huxley Huxley and Simmons It was built using a reductionist approach that takes into account the actual molecular structure of the muscle and attempts to predict the developed tension by simulating the forces produced by the crossbridge attachments between the actin and myosin molecules.
When building a biomechanical simulation of muscle a few key goals must be kept in mind. The primary goal is that the muscle model generates outputs that are a reasonable reproduction of what a real muscle would do in a similar situation. The biomechanical model does not have to produce the exact behavior, but it should be close within some well-defined range of error. Also, it should never produce results that are far outside the biologically realistic range of outputs for any situation, and it must be capable of functioning over the entire operating range of the real muscle.
The model should also be as simple and fast as is feasible while maintaining biological realism. Finally, when choosing which model to use for the simulation it is important to keep in mind the type of data that is being sought.
The Huxley crossbridge models are very good at addressing questions related to the intricate internal details of what is actually happening within the muscle such as how much energy is being used, and of taking into account a number of properties seen in muscles that are not reproduced in other models.
However, this level of detail is offset by the enormously greater complexity and computational requirements needed for these types of models. This is the major reason that Hill type systems are still the predominant model used in biomechanical simulations of multi-joint systems.
The Hill model fails to provide any data on the internal details of the muscle, but in simulations such as this, that type of data is typically not important. The overall goal is to use multiple simulated muscles that can produce torques around joints to reproduce biologically realistic movements.
For these types of simulation it is only the force output relationship that is important, and Hill models are more than adequate to provide this type of data. Muscle Model Figure 1. There are some differences, but these will be discussed below.
The hill muscle model used for the biomechanical simulations within AnimatLab is diagramed in figure 2. There is a parallel spring that simulates the passive elastic components of the muscle.
This elasticity is primarily produced by the connective tissues within the muscle Winters There is another spring in series with a contractile element. In part A of figure 2 the contractile element is seen as a dashed box with a force generator and a dashpot within it.
Sarcomere length-tension relationship
A dashpot is an element that provides viscous friction that opposes motion. There are three primary relationships that are important to emulate the behavior of muscle.
These are the tension-length, force-velocity, and stimulus-tension curves. We will talk about each of these, and how they are implemented in AnimatLab, below. Stimulus-Tension Relationship Figure 2. Adapted from Shadmehr and Arbib When motor neurons fire they release acetylcholine at the neuromuscular junctions that depolarizes the sarcomela. This wave of depolarization rushes through the T-tubule system to the sarcoplasmic reticulum where calcium ions are released and act on the myofibrils to elicit contractions Becker, Kleinsmith et al.
An impulse stimulation produces an isolated twitch shown as the single bump at 1 Hz in figure 3. As the motor neuron fires faster these twitches begin to merge together until finally they reach some maximum value and the muscle is said to have reached tetanus, or maximal activation.
If one plots the maximum tension that can be developed at each stimulus frequency a sigmoidal curve such as the one shown in part b of figure 3 can be produced that relates the strength of the stimulus to the maximum tension.
Typically there will be multiple motor neurons involved in stimulating a given muscle. The more of these neurons that are firing the more fibers in the muscle are recruited and the larger the force that is produced. This behavior is not explicitly modeled in AnimatLab. This is one of the biggest difference between the Shadmehr and Wise; Shadmehr and Arbib muscle implementation and the one in AnimatLab.
They explicitly model calcium activation of the muscle. Instead, in AnimatLab a non-spiking neuron is used to emulate the membrane of the muscle. Motor neurons connect to the non-spiking neuron to excite or inhibit it. The time constant of that neuron then determines how those inputs are integrated. This is related to force using an adapter.