### Mathematical Modeling in Your Bone

Are you studying math or physics and would be interested to know some aspects of interdisciplinary research in math/physics and biology/physiology? Are you curious about the picture above? If you answered “yes” to either of the questions, read on!

Having just finished my first year undergraduate studies in Physiology and Mathematics, I will explain how mathematical modeling is used in my research project with Dr. Komarova, McGill University, Dept. of Dentistry, now also collaborating with Shriners Hospital for Children.

This project is on osteoblasts, cells that produce bones, balancing out the bone resorption by osteoclasts. In our experiments, the research team wishes to determine how osteoblasts respond to external mechanical stimuli and communicate with each other. We used Atomic Force Microscopy (AFM)—essentially a needle with a tip of atomic scale—to deform the osteoblast. To analyse the osteoblast’s response, we examined its release of calcium, a common cell-signaling chemical. It was found that after the initial deformation by AFM, the calcium level of both the primary cell (the cell we deformed) and its neighbouring cells increased. This suggests that osteoblasts can communicate with each other, though it is not clear why. Since ATP is a common intercellular signaling mediator, we are studying the various ATP receptors on the osteoblast. We try to understand how osteoblasts could use ATP for intercellular communication and cause their neighbour cells to increase their calcium levels as well. My job in this project is to process the data on calcium release in response to different concentrations of ATP ejected in a group of osteoblasts.

Now how has mathematical modeling been used in this project?

First, modeling helps simplify the relationship between receptors and their respective signaling chemicals quantitatively. As (1) there are many receptors in a cell, (2) each receptor may respond to more than one agent, (3) not all respond to ATP, and (4) more than one receptor can respond to ATP, understanding these receptors based purely on their biochemistry may not be sufficient. Because the activation of receptors often depends on the concentration of the agent (in our case, ATP), modeling can be very powerful. In this study, modeling can help understand how the activation of one receptor is affected by the activation of another (i.e. how the receptors interact in the network), as well as how this network depends on ATP concentration.

My graduate student supervisor, Stella Xing, has previously performed experiments on the effect of various ATP concentrations on the activation of the six ATP receptors (P2X and P2Y families) in osteoblasts. She then found the mathematical functions for each receptor using ATP concentration as the variable.

My work is in the second application of mathematical modeling in the project. Remember that we are studying the osteoblasts—ultimately, we are interested in how their intercellular signaling is controlled. To this end, another set of data on the effects of various ATP concentrations on osteoblast calcium levels was gathered. I am responsible for combining them into one, and use this one function to mathematically fit osteoblasts’ calcium release data. In other words, we are attempting to understand how controlled activation of each P2X/P2Y receptor in osteoblasts affect their signaling in response to external mechanical stimulation. The fitting of the equation to experimental data requires more than one try, and this part is where the art of modeling lies. If we only fit the data with the combined function of the six receptors all at once, then we will get a fit like this:

This is obviously a poor fit. To find a better fit, we must divide the data into sections, and fit only the activated receptors for each section:

We fix the values found for each section, and then fit the remaining data by looking at all the data together, eventually finding the coefficients for all six receptors. The coefficients can reveal information such as whether the receptor is inhibited (a negative value) or activated (a positive value), and how strongly it is activated/inhibited. The change of coefficients when fitted with more receptors could suggest how the receptors may be affected by the activation of others.

Stella has previously found the best fit for one cell, I am now analyzing the group of osteoblasts contained in one field of view. We hope to determine a numerical coefficient for each ATP receptor’s activation equation in the combined function, that explains both one cell and multiple cell results. This would allow us to better understand why such complex networks of receptors exist and may be required in osteoblasts, as well as how they cooperate (activation/inhibition) with all other receptors. This type of quantitative description of biology can provide a simple guideline into more complex biochemical, molecular, physiological research.

Modeling is not limited to such straightforward application. It could also help to eliminate the “noise” (experimental artifacts, deviations in biological systems, etc.) in experiments, another interesting use to be explored another day.

Editing LH.

## Leave a Reply

You must be logged in to post a comment.