HHMI BSQB Team

 

Overview

 

This is the home page of HHMI BSQB Team. The purpose of this team is to figure out some proper and useful materials for undergraduate students both in math and biology to study.

 Group Members

 

• Zunlei Xiao (xiao@math.udel.edu)
• John A. Pelesko (pelesko@math.udel.edu)

 

1. Introduction

 

(1) Biological Problems

 

 We started with a particular subject based on the PhD research of Brian Danysh, which centers on the study of lens capsule, a contiguous basement membrane surrounding the lens of the eye. This is also the Delaware Problem in MPI 2007.

 

The lens is a transparent component of the eye which focuses light rays entering the eye to appropriate places at the back of the eye. It is tendered by muscles to the surrounding tissue and it is the movement of these muscles that alters the shape of the lens and alows focusing at different distances. The lens is sandwiched between two aqueous compartments: the aqueous humor in front of the lens and the vitreous humor behind. The lens is an avascular tissue and relies on diffusion of nutrients and waste products for survival. It consists of highly specialized transparent cells surrounded by an outer membrane capsule. The outer membrane is a porous scaffold to which the cells underneath the membrane attach. The capsule allows force transmission and is selectively permeable. It is a complete seal around the lens and has thickness of 26 m in a human and 10 m in a mouse. There are various pore sizes (99% are 4-5 nm and 1% are 10 nm) and these are small enough to prevent cells and some molecules from entering the layer. The scaffold is made from Collagen 4 and has sugar molecules bound on which extend into the pores and "wave in the wind".

 

We are interested in studying diffusion through the lens capsule because diffusion is important for lens development and growth, nutrient and waste release, drug delivery, ocular inflammation and for cataract formation and treatment. To determine the diffusivity of various molecules through the lens capsule, experiments have been carried out using a technique known as fluorescence recovery after photobleaching (FRAP). In this process, the lens is immersed in a bath of fluorescent molecules and left to soak for more than an hour, which allows the molecules to diffuse into the lens capsule and to ensure that the whole system is in chemical and diffusional equilibrium. Some of the molecules within the scaffold remain free to diffuse around in the medium and some become bound to the scaffold. The proportion in each of these "compartments", and the affinity of the molecules for the scaffold, depends on their chemistry (e.g. size, charge, etc).

 

A high-intensity blue laser is used to bleach out a 5 m radius tunnel through the capsule, creating a region of interest (ROI). The laser is focused at a plane within the capsule and, while the bleaching is most effective at this plane, bleaching occurs throughout the tunnel. After 250 ms, the laser is turned down to a very low intensity and used to take photographs of the focal circle. After the bleaching, a proportion of the cells within the 5 m circle have stopped fluorescing, and the photographs show how the intensity increases with time as molecules from outside the circle diffuse in. The average intensity in the circle is calculated by counting up the pixels in the circle and dividing by the area. However, the amount of flourescence decreases with time, so the fluorescence is also calculated over an identical "unbleached" region 50 m away from the ROI, and the data in the ROI is normalized by dividing by these data.

 

Currently, the post-experiment data processing involves the following recipe. First, the normalized data is plotted and fitted with either

 

     or    , (1.1)

 

where they treal all the parameters in (1.1) as independent. Unsurprisingly, the "double exponential fit" with 5 independent fitting parameters fits the data better than the "single exponential" with 3 parameters. After has been determined, the diffusivity is calculated using

 (ref i.b), where is the radius of the region of interest and is a correction coefficient. There are two types of behavior exhibited by the kind of molecules used in the experiments, as illustrated below.

 

 

 

 

 

 

 

In Figure 1.3, the intensity curve tends to a steady state over the time scale of the experiment. This graph is associated with molecules that are tightly bound to the scaffold (and cannot leave).In Figure 1.4, after the initial exponential transient, the intensity slowly increases back to its original value. This graph is associated with experiments where the bound molecules are exchanging with unbound ones.

 

The experimentists are concerned with the procedures they use to interpret the data. Firstly, they are worried about the fact that they neglect all the influence of the third dimension, assuming that diffusion occurs only in a plane with no z-axis diffusion. Secondly, they are worried about whether the bleaching creates an aurora at the edge of the ROI which would alter their results. Thirdly, they are worried about when to truncate their time series and to fit the data: they find that truncating their data 10 seconds later can alter the fit parameters by 10%. So we need to tackle the following questions:

 

(a) Why does the double exponential fit better than the single exponential?

(b) Why does the time at which you terminate the data alter the predictions?

(c) Is there a more robust way to fit the data than is currently being used?

(d) Does the effect of transfer between the mobile molecules and the bound ones explain the difference between the two types of graphs?

 

 

 (ref. a)

 

References

 

a. C. Breward, D. Edwards, M. Haider, J. Pelesko, G. Schleiniger, MPI 2007 Draft, Edition of July 13, 2007

 

i. What is FRAP

 

The technique of FLUORESCENCE RECOVERY after PHOTOBLEACHING (FRAP, also called fluoresence photobleach recovery) was introduced in 1970s to study the diffusion of biomolecules in living cells (Edidin et al. 1976). FRAP denotes a method for measuring two-dimensional lateral mobility of fluorescent particles, for example, the motion of fluorescently labeled molecules in ~10 region of a single cell surface. A small spot on the flourescent surface is photobleached by a brief exposure to an intense focused laser beam, and the subsequent recovery of the fluoresence is monitored by the same, but attenuated, laser beam. Recovery occurs by replenishment of intact fluorophore in the bleached spot by lateral transport from the surrounding surface. (ref b.) For several years, it was used mainly by a small number of biophysicists who had developed their own photobleaching systems. Since the mid 1990s, FRAP has gained increasing popularity, due to the conjunction of two factors. First, photobleaching techniques are easily implemented on confocal laser-scanning microscopes (CLSMs) (McNally and Smith 2001). FRAP has therefore become available to anyone who has access to such a microscope. Second, the advent of the green fluorescent protein (GFP) has allowed easy fluoresent tagging of proteins and observation in living cells.  (ref a.)

 

 

References

 

a. G. Rabut, J. Ellenberg, Photobleaching techniques to study mobility and molecular dynamics of proteins in live cells: FRAP, iFRAP, and FLIP, Gene Expression and Cell Biology Programmes, European Molecular Biology Laboratory, D-69117 Heidelberg, Germany

b. D. Axelrod, D. E. Koppel, J. Schlessinger, W. W. Webb, Mobility measurement by analysis of fluorescence photobleaching recovery kinetics, Biophysical Journal Vol. 16 pp. 1055-1069, 1976.

 

 

ii. Why Studying FRAP

 

 

 

iii. Who Care in FRAP

 

 

 

(2) Mathematical Models & Techniques

 

i. diffusion equations, initial & boundary value problems

ii. separation of variables

iii. bessel functions

iv. parameters fitting

 

 

2. Simple Model

 

 In this section, we will set up a simple one dimensional model to have some basic idea of what is going on.

 

At the beginning of the experiment, the membrane is satureted with fluorescing particles, and allowed to come to equilibrium. Then bleaching occurs in the ROI, where we consider to be a line segment . We consider the bleaching to be rapidly symmetric. After bleaching, the measurement phase begins. We take  to be the time at which bleaching occurs. Therefore, we consider the end of the preparation phase to be .

 

 The fluorescing particles come in two varieties: immobilized molecules which have attached to the membrane (denoted by ), which do not diffuse, and mobile molecules (denoted by ). By "equilibrium" above, we first mean transport equilibrium, which implies that the particles are uniformly distributed throughout the membrane:

 

 

where are constants that can be measure at , and we have chosen the spatial variable to be .

 

We also mean the "chemical equilibrium" of the mobile and immobile species. We assume that there are enough binding sites so that the depletion is never an issue. Since the immobile species cannot diffuse, we see that its evolution equation is given by

 

  (2.0)

 

where is the rate constant for the association kinetics, and is the rate constant for the dissociation kinetics. Since the preparation phase has been allowed to proceed to steady state, we have the following:

 

 

Here is called the affinity constant. Combining the equations above, we obtain,

 

      (2.1)

 

Then at the very beginning of the measurement phase, a laser beam bleaches some fraction of the fluorescent particles inside the ROI, permanently removing their fluorescent properties. For the purposes of this section, we assume that

(1) The bleaching process is uniform in space, bleaching a constant fraction of the fluorescent molecules. (In Brian's experiment, .)

(2) The bleaching process is taken to be faster than any other process in the problem, so it may be considered as instantaneous. Those bleached particles will be denoted by .

 

Then from (2.1) we see that initially we have the following in the region of interest:

 

  (2.2)

 

Note that we are changing only the labelling; the transport and chemical properties are unaffected. Therefore, (2.1) holds for all as long as we consider the immobile or mobile molecules together as a group:

 

   (2.3)

The light intensity is proportional to the number of fluorescent molecules, so we can use intensity and consentration interchangeably. The measured intensity is simply the average over the ROI of all the fluorescing species : 

 

Substituting (2.1) and (2.2), we have

 

 

To make it simpler, we begin with the case where kinetics are unimportant. That is, there is no interchange between the immobile molecules and the mobile ones. Without kinetics, the equation governing is just the diffusion equation

,  (2.4)

where is the diffusion coefficient. In order to solve the problem, we need boundary conditions on the various species. Since we have assumed the bleaching symmetric, we may specify at some distance , the lens will transport any needed particles to and from the boundary very quickly. Therefore, we may specify Dirichlet data there that is specified by the preparation phase in (2.1), since we have outside the bleaching zone:

 

 

 

To make it easier, let's now assume , which means, we assume this is a one-compartment model. So we have the boundary condition

 

  (2.5)

 

and the initial condition is

 

  (2.6)

 

So we have established a one dimensional model. In such case, remains at its initial value, so the intensity becomes

 

 

Now let's simplify this diffusion equation. First, we do the variable transform

,

then the diffusion equation (2.4) with the initial and boundary conditions can be written as

 

  (2.7)

  (2.8)

  (2.9)

 

To transform to homogeneous end condition, we let

 

 

and then subsitute this expression into (2.7) - (2.9) to obtain

 

(2.10a)

(2.10b)

 

 Now let's solve this initial boundary value problem. Separating variables by setting , we have the following

 

 

that is,

.   (2.11)

 

The left side of (2.11) is a function of , while the right side is a function of , and they are independent variables. This guarantees that both sides must be a fixed constant . Hence we can write

 

 

or

 

  (2.12)

 

So now we can solve each of these ODEs, multiply them together to get a solution to the PDE. Both equations are standard-type ODEs and have solutions

 

  (2.13)

 

where are constants. Hence all functions

 

 

will satisfies the diffusion equation. (Note that here is different from the one in (2.13).) Now let's find the solution for (2.10). Plugging the boundary conditons, we have

 

 

Since is not constant zero, at least one of is nonzero. Without loss of generality, let's assume Then

 

 

This implies

 

Therefore, for each , there is a with corresponding characteristic function such that 

 

 

satisfies the boundary conditions. Let be such that

 

 

where is to be defined later. Now let's write

 

  (2.14)

 

Apparently, this satisfies our diffusion equation and the boundary conditions. We only need to find . Plugging (2.14) into the diffusion equation and the initial problem, by the completeness of the characteristic functions , for each , we obtain the ODE with initial condition:

 

 

Solving the ODE we obtain

 

 

We now have the solution to our original initial boundary value problem

 

 

Plugging the relation and , we obtain

 

   (2.15)

Note from (2.15) that , as expected. Therefore, we have that

 

 

which is actually,

 

   (2.16)

 

(2.15) and (2.16) are the solution to this simple model.

 

Suggested Problem: Use Fourier or Laplace Transform to solve the problem, if possible.

 

 

Reference

 

a. Stanley J. Farlow, Partial Differential Equations for Scientists and Engineers, Chapter 2, Dover, 1993

b. C. Breward, D. Edwards, M. Haider, J. Pelesko, G. Schleiniger, MPI 2007 Draft, Edition of July 13, 2007

c. Lishang Jiang, Yajian Chen, Xihuan Liu, Fahuai Yi, Notes on Mathematical Physical methods, (Chinese Edition), pp 150-161, Higher Education Press, 1996.

 

3. Better Version

 

In the simple model above, we have made a series of assumptions, for example, one-dimensional, one compartment, no kinetics etc, most of which can be released to get much better model in this chapter.

 

(1) Two-dimensional one compartment model without kinetics

 

As we mentioned above, real human lens is a membrane, a thin, pliable sheet. It should be treated as a at least two dimensional manifold, instead of a line segment, to get the most accurate description of the diffusion process of the fluorescent species after bleaching. Let's first ignore the thickness of the lens and consider the lens is a two dimensional surface. This assumption is valid as the lens is very thin. Still, to make our life easier, first let's consider the one compartment model without kinetics.

 

There are only a few changes with the experimental setup as in section 2. Conditions (2.1)-(2.3) can be rewritten as

 

  (3.1)

 

  (3.2)

 

  (3.3)

And the intensity is

 

  (3.4)

 

where we have exploited the radial symmetry of the problem. Substituting (3.1) and (3.2) into (3.4), we have

 

  (3.5)

 

We are dealing with the two dimensional model, so by changing to the polar coordinate, the diffusion equation is

 

  (3.6)

where  represents for the radius. In order to solve the problem, we need boundary conditions on the various species. As in the section 2, we may assume there exists some distance , such that

 

  (3.7)

 

Since this is a one compartment model,  holds. By using the scaling

 

,

 

we transform (3.6) and (3.7) into

 

  (3.8)

      (3.9)

 

The ROI now becomes , and (3.4) becomes

 

    (3.10)

 

In this case,  remains at its initial value given in (3.2), so

 

    (3.11)

 

To transform to homogeneous end conditions, following the similar procedure as one dimensional case, we let

 

 

and then substitute this expression into (3.8), (3.9) and (3.2) to obtain

 

  (3.12a)

 

  (3.12b)

 

Separating variables by setting , we have the following:

 

  (3.13) 

 

So we get

 

   (3.14)

 

where  is the th zero of Bessel function , . Then solving the  equation, we obtain

 

 

  (3.15)

 

Then by the principle of superposition, we have that

 

  (3.16)

 

where by the orthogonality of the Bessel function we have the following

 

   (3.17a)

 

   (3.17b)

 

where the use of the argument 1 for  will become clear below. The relationship for the norm of  is given by Theorem 4.23 in Bell, while the integral in  is given by Theorem 4.8(i) in Bell.

Substitiuting (3.17a) into (3.16) and using the transform again, we obtain

 

    (3.18a)

 

  (3.18b)

 

Note from (3.18b)  as expected. Therefore, we have

 

 

 

  (3.19) 

 

where we have used (3.11). By substituting (3.17b), we obtain

 

  (3.20)

 

Therefore, (3.18b) and (3.20) are the solution to this two dimensional, one compartment without kinetics model.  

 

 

 

 

 

(2) Two-dimensional two compartment model without kinetics

 

In some sense, the assumption that  is suspect. Since there is no change in the medium at the bleaching zone interface, there is no reason to suspect that we should impose a Dirichlet condition there. Therefore, we may introduce a two-compartment model by defining

 

and not requiring that . (See Figure 3.1.) We here have two problems dealing with the two component model. First, in this case,  is no longer a constant. If we still take the uniform bleaching within the zone, we have

 

   (3.21)

where  is the Heaviside step function. Thus (3.17a) becomes

 

  (3.22)

 

where

 

  (3.23)

 

Plugging (3.22) and (3.23) into (3.18b), we obtain 

 

  (3.24) 

 

Second, in this case, the sensing is not uniform, either. The signal strength  varies with . Therefore, we should modify the definition of intensity to

 

  (3.25)

 

Here  is a normalization constant chosen so that averages of constants remain constant. Thus the integral in (3.19) should be replaced by

 

  (3.26) 

 

If we average uniformly within the zone, we obtain

 

 

  (3.27a)

 

  (3.27b)

 

Thus (3.19) becomes  

 

  (3.28)

 

Therefore, (3.24) and (3.28) give the solution to the two dimensional, two compartment without kinetics model case.  

 

Note that if , which corresponds to the one compartment model, (3.24) and (3.28) reduce to (3.18b) and (3.20), respectively.

 

 

 

(3) Two-dimensional One compartment with kinetics

 

Now we treat the case with kinetics. To make it simpler, still, we consider one compartment situation. In this case, the amount of  can by changed by binding or dissociation, so (3.8) is replaced by

 

  (3.29)

 

which we wish to solve in tandem with (2.0). We want homogeneous end conditions for both dependent variables, so we let

 

  (3.30a)

  (3.30b)

 

Substituting (3.30a) and (3.30b) into (3.29) we obtain

 

  (3.31a)

 

where we have used the fact that  holds for the bulk values. Similarly, (2.0) becomes

 

  (3.31b)

 

and the boundary conditions for  become

 

  (3.32)

 

The only thing we need to do is to solve the initial boundary value problem (3.31a), (3.31b) and (3.32). Motivated by (3.18a), we let

 

 

In particular, we see that  

 

  (3.34)

 

Simplifying the above and extending it to the other species, we obtain

 

  (3.35)

 

Substituting into (3.31a) and (3.31b), we obtain

 

  (3.36a)

  (3.36b)

 

where we have used (3.14). Note that since the equations are homogeneous, all the odd coefficients cancel from each term. Rewriting the above in matrix form, we have

 

   (3.37a)

   (3.37b)

 

where   are the eigenvalue-eigenvector pairs of , given by

 

   (3.38)

 

The initial conditions for the problem are given by

 

 

Solving this equation for the , we obtain

 

  (3.39a)

  (3.39b)

 

Note that as the rate constants go to zero, we have

 

 

Here we reduce to the no-kinetics case where is a constant and decays as .

Substituting our expressions into (3.10) to obtain the intensity, we obtain the following

 

  (3.40)

 

 

Suggested Problem: Following the same procedure, establish models for one dimensional two compartment without kinetics case and one dimensional one compartment model with kinetics case.

 

 

Reference

 

a. C. Breward, D. Edwards, M. Haider, J. Pelesko, G. Schleiniger, MPI 2007 Draft, Edition of July 13, 2007

b. W. W. Bell, Special Functions for Scientists and Engineers. London: D. van Nostrand, 1968