Biosensors

Title: Biosensors: Cell Communication, Experimentation, and Modeling

Authors: Rob Carr, Marc Palmer, Tommy DiRaimondo, Matt Pickvet

Date Presented: September 19, 2006 Date Revised: September 26, 2006


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Introduction
Cancer cells, erectile dysfunction, and your body's growth are all directly related to biosensors. Every cell contains receptors, which, when bonded with an appropriate ligand (small biological molecule) or protein (biological macromolecule), will send out a signal to trigger some other function of the cell. This concept is important for chemical engineers because they are often consulted to assist with finding the rates at which ligand/receptor pairs bind. The rate constants for those reactions must be determined in the midst of all other cell activities, which is quite a challenging task. Biosensors can be thought of as control devices for a chemical processing plant. The control devices are nanometers in size and the chemical plant is the human body. It is easy to draw analogies between biosensors and real life mechanical process sensors because both essentially monitor, change, and control cells in the human body or valves and other control devices respectively. For example, a receptor protein on the surface of the cell can open and close ion-channel on the cell. Similarly, a flow controller can control an actuator on a valve causing the valve to open or close. Both control devices, biological and mechanical, serve the same purpose but in different environments and on very different scales.

Background
The mechanisms of cellular communication stretch across all spans of life. Signaling between cells is an essential tool in cellular development. The complexity of multicellular organisms demands on a highly efficient network of signaling between molecules. These signals guide the cell through its development from differentiation all the way through to the destruction of the cell.

Mechanism of Cell Signaling
One of the keystone elements of cellular communication is extracellular signal molecules. These molecules are produced by the cell to communicate to other local cells. The signal molecules consist of, but aren’t limited to, proteins, peptides, amino acids, nucleotides, steroids, and dissolved gases. They are typically sent out of the cell either by exocytosis (an elaborate process in which the cell excretes signal molecules through it’s membrane), diffusing directly through the membrane, or simply binding to the extracellular surface of the cell.

Once the signal molecule leaves the host cell, its second objective is to seek out the corresponding target cells and specifically bind with a particular protein called the receptor. The receptor is typically a transmembrane protein (stretched across the entire membrane) located on the surface of the target cell. However, the receptor could be located within the target cell, which requires that the signal molecule be small and hydrophobic in order to diffuse directly through the plasma membrane of the cell. Once the signal molecule binds to the receptor, the receptor becomes active and sends a cascade of signals throughout the target cell, which in turn alters the cell's biomechanics.

Types of Communication
There are two major sub-divisions of cellular communication: extracellular and intracellular. Both of these involve the activation of proteins via the binding of signal molecules with receptors; the two only differ by the fact that extracellular communication takes place outside of the cell, while intracellular takes place inside of the cell. However, the two are not independent of one another. They can be thought of as stages of communication between cells; the first stage being extracellular communication, which binds signals to the target cells, which then triggers the second stage - intracellular communication. After the onset of intracellular communication, a cascade of interactions changes the behavior of the target cell internally.

Local Extracellular Communication
As discussed above there are several instances where the signal molecules can remain bound to the surface of the signaling-cell and induce target cells that only come into direct contact with it. This type of signaling is termed contact-dependent signaling and is crucial in processes such as development and immune responses. In most cases the signal molecules are secreted away from the cell and seek out their target cell via diffusion through a media. The distance that the signal molecules diffuse from the signaling cell to the target cell varies depending on the type of cell. Signaling molecules that travel short distances act as a local mediator and only influence cells in their immediate area; such a process is termed paracrine signaling and requires that the signaling molecules be rapidly taken up by local cells or risk being destroyed by extracellular enzymes (An enzyme is a catalytic protein). This ensures that signals are only received by their immediate neighboring cells.

Cells can also send signals that only affect neighboring cells of the same type, including themselves. This process is called autocrine signaling and is crucial during the development stage of an organism when the cells begin to differentiate. The cells send these signals to themselves to support this developmental specialization. Autocrine signaling is amplified when it is performed simultaneously with local cells of the same type, and acts as a reinforcement for groups of cells to make the same developmental selection.

Neighboring cells can also communicate via specialized cell to cell interactions where the two cells' plasma membranes overlap and connect directly via cytoplasmic regions; these multi-cellular connectors are termed gap junctions (Refer to example problem 1). The molecules that can pass through these junctions are limited to smaller molecules such as calcium ions or cyclic AMP; proteins and peptides are typically too large to pass through. Gap junctions allow for direct communication between cells without having to pass through either cells' plasma membranes.

Long Distance Extracellular Communication
Paracrine secretion is often insufficient for more intricate organisms. These organisms require signaling that may span across several different parts of the body. A good example of long distance communication within the body is synaptic signaling. This is accomplished by nerve cells that when activated by signals from their environment, send electrical impulses rapidly down their axon (a long biological shaft extending far distances). When the signal reaches the end of the axon, nerve terminals release a chemical called a neurotransmitter through a specialized junction called a chemical synapse. These synapses then ensure that the neurotransmitter reaches a specific postsynaptic cell. Endocrine cells also release signals controlling the body. These signal molecules, called hormones, are emitted from the cell into the blood stream. In contrast to synaptic signaling, endocrine signaling is much slower and takes place at a much lower concentration. Synaptic signaling involves electrical impulses that can travel long distances almost instantly ending in the release of signal molecules that only need to diffuse a very short distance to reach the target cell. However, endocrine signaling requires a larger diffusion through the blood stream in order to reach the target cell and is therefore much slower.



Extracellular Response of Non-Similar Cells
The way in which a cell reacts to its environment varies from cell to cell. There are several factors that attribute to this: differences in receptor proteins that determines the amount of signals to which the protein can respond, and differences in intracellular machinery which the cell uses to interpret the signals. In many cases, the different cells may have the exact same receptor protein that receives the same signal molecule, but produce very different responses due to differences in the intermolecular machinery of the cell.

Turnover Rate of Effected Molecules
The effect of a signal can be very long lasting during the development of an organism—some of them indefinitely. However, as an organism gets older the response to a certain signal fades as the signal is removed. Therefore, it is important to consider the process when a signal is released from the receptor. For example, when new signals are received and the old signals are released, all traces of the old signals' effects are erased. Therefore, due to the rapid turnover of the effected molecules, the signals' effects are short-lived. Hence, the rate of turnover and destruction of the effected molecules are crucial to cellular communication.

Direct Diffusion Through Plasma Membranes
As discussed earlier, some small and hydrophobic signal molecules can diffuse through the plasma membrane and directly influence the activity of an intracellular protein.(Refer to example problem 1) For example, dissolved gases such as nitric oxide (NO) can dissolve through the membrane of muscle cells and signal them to relax. This principal is the fundamental basis for medication that treats erectile dysfunction. Nitric oxide enters local blood vessels of the penis and causes the blood vessels to dilate inducing an erection.

There are a large number of small hydrophobic signaling molecules that diffuse directly across the plasma membrane and bind to intracellular receptors, such as: steroid hormones, thyroid hormones, retinoid, and vitamin D. When these signal molecules bind to their receptors inside the cell, they activate proteins that in turn bind to the DNA to alter the transcription of specific genes. The receptor proteins that alter DNA are all part of the nuclear receptor family (NRF). Many times when receptors from the NRF attach to DNA they can trigger the transcription of a gene that in turn activates a transcription of another gene and so on. Therefore, the presence of one receptor in the NRF can cause an array of complex gene expressions.

Cell Signaling Pathways
In general, cells send and receive signal molecules that influence cell behavior. When a cell binds a signal molecule (ligand) using receptor proteins embedded in its plasma membrane, the signal must be relayed to the interior of the cell in a uniquely designed signal pathway. This signaling pathway takes a complex form, but the idea is simple: A signal molecule on the outside of the cell is bound by a receptor that in turn relays the signal through multiple proteins inside the cell until the target protein is reached. Once the target protein receives the message, the protein changes and therefore alters the cell’s behavior.

Cell Surface Receptors
The three main classes of cell surface receptors are ion-channel-linked receptors, G-protein-linked receptors, and enzyme-linked receptors. These types of receptors bind signal molecules, including neurotransmitters and signal proteins, on their surface. This binding of a ligand is then converted into intracellular signals used to alter the behavior of the target cell.

Ion-channel-linked receptors
Ion-channel-linked receptors are involved in synaptic events where neurons send the signal to the target cell. Once the signal (neurotransmitter) binds to the receptor protein, the receptor itself acts as an opening and closing channel allowing other molecules to enter and exit the target cell.

G-protein-linked receptors
G-protein-linked receptors bind a signal molecule and indirectly affect a separate protein, either an enzyme or ion-channel. Once the G-protein-linked receptor binds a ligand, the receptor activates a protein, called a trimeric GTP binding protein, that in turn activates either an enzyme or an ion channel-protein.

Enzyme-linked receptors
Enzyme-linked receptors act either as the enzyme themselves or as the protein that activates an enzyme. Once the enzyme-linked receptor binds a ligand, it either becomes an activated enzyme itself, or a protein that serves to activate a different enzyme.

Relay Mechanism from the Surface to Interior
Signals or ligands bound by either G-protein-linked receptors or enzyme-linked receptors are transmitted to the interior of the target cell through other small and large signaling molecules. Once the surface receptor binds a ligand on the surface of the cell, that signal is translated to the interior of the cell where small and large messenger molecules are made. These messengers take the form of small signals, similar to the ones the surface receptor binds, or larger signaling proteins. The small signals diffuse through the interior of the cell away from their source due to concentration gradients. The larger signaling proteins relay their message by activating other proteins or generating small signals themselves. These signals in the interior of the cell are then free to affect target proteins, which in turn alter the behavior of the cell itself.

Molecular Switches
Many of the above intracellular signaling proteins act as molecular switches, where they activate or deactivate due to binding of a signal themselves. The largest class of these proteins are activated and deactivated through phosphorylation. Phosphorylation occurs when either a protein kinase adds phosphorus groups or a protein phosphatase removes phosphorus groups, activating and deactivating the intracellular signaling protein respectively. The second main class of molecular switches are GTP-binding proteins. These proteins are activated through the binding of GTP and deactivated through the binding of GDP. Both molecular switches are important in the trafficking of signals inside the target cell.

Multiple Intracellular Signal Interpretation
It is almost always the case that a single target cell contains hundreds of signals at the same time and must decide what it needs to do. The cell accomplishes this task through integrator proteins located in the cell’s interior. These integrator proteins receive multiple signals and create an output that causes the desired change in the behavior of the target cell.

Acceleration of Intracellular Response
Inside the cell there are hundreds of different signaling pathways being utilized at the same time. In order to relay the signals quickly and efficiently, mixing and tangling of separate signal pathways must be avoided. This task is accomplished by scaffolding proteins, which keeps the signaling pathways organized. The scaffold proteins guide successive signaling steps in the signaling pathways by grouping the appropriate signaling proteins into a larger complex. This inherently speeds up the signaling pathway and in turn the target cell response.

Evolution of Signaling Pathways
Inside a cell it is possible for random protein signaling complexes to form through small binding domains. Due to these binding domains, different arrangements of signaling proteins can come together to form an entirely new protein complex. These protein complexes are then responsible for the determination of the routes for signaling pathways. Furthermore, new combinations of signaling proteins create new signaling pathways. The key to this form of rapid evolution is the binding domains that serve as docking sites for different combinations of signaling proteins.

Cell Response to Signal Concentration
Often it is found that the response time of a cell is proportional to the concentration of signal molecules designed to provoke that particular cell response. This means that the cell responds gradually to the changes in the signal concentration. However, it is sometimes found that many cell responses initiate suddenly. Some responses are even undetectable until a threshold concentration of signals is reached at which time the cell rapidly responds. The latter case of a threshold concentration can be explained by a positive feedback mechanism. In this case signals activate an enzyme that then produce more signals that serve to activate the enzyme further. Once the signals reach a threshold value, the enzyme self-accelerates and the cell response is almost immediate.

Visual Aid
Here is a picture that summarizes the pathway for a cellular communication system.

Worked out Example 1
For the following signaling pathways labeled A,B, & C describe the process and the key components for each.

Solutions: Pathway A: Pathway A illustrated above, is a simple gap junction that allows for the free direct passage of small molecules from neighboring cells' cytoplasms without traveling through the plasma membrane.

Pathway B: Pathway B illustrates contact dependent signaling between a surface bound protein and a G-protein-linked surface receptor. Once the surface bound protein and G-protein-linked receptor contact, the complex formed activates a GTP-binding protein inside the cell which in turn activates either an enzyme or an ion-channel.

Pathway C: Pathway C illustrates a small signal molecule diffusing directly through the plasma membrane. This signal binds and activates an intracellular receptor protein forming a messenger protein complex that interacts with DNA to regulate transcription. These regulating protein complexes are part of the greater nuclear receptor superfamily.

Experimentation of Receptor Binding
When studying biosensors, it is important to explore how different molecules work together, specifically, the binding of ligands (molecules usually in the form of hormones or antibodies) to cell receptors. This combination is important because a receptor sends out a signal to trigger another process once it is bound to a specific ligand, resulting in various intracellular based behaviors such as growth and protein synthesis. Since ligand-receptor reactions are relatively easy to track with regards to cell activities, reaction rates and concentrations can be determined. Engineering skills are needed to determine the rate constants and concentrations of ligands binding with receptors while also taking into account receptor mobility and trafficking events of the complex - both of which can significantly affect the results of an experiment. This is a common application for determining drug release times in medicines. As seen in the following sections, this type of information is generally found in laboratories at the microscopic level.

Tracking Ligands
Engineers and biologists combined have developed five ways to help distinguish the binding of ligands from all of the other trafficking activities possibly occurring during experiments, such as degradation, synthesis, and recycling. One should use as many of the above conditions as possible when conducting an experiment. These methods are used to simplify the model in reducing the number of variables. However, they are not to be assumed to produce overly accurate results within the human body due to the temperature and pH varying insignificantly.
 * 1) Perform the experiments in the presence of trafficking inhibitors.
 * 2) Isolate the membranes being used in the study.
 * 3) Perform the experiments at temperatures below room temperature and above freezing temperatures.
 * 4) Use engineering models to estimate all rate constants of the binding reactions.
 * 5) Use procedures which will essentially separate the complexes from trafficking (a pH above 7).

As mentioned above, ligand/receptor complexes are modeled because of the ease of tracking ligands’ movements. The two most common methods of doing this are by radioactively or fluorescently labeling the ligands before the experiment is begun. In order to track the binding of ligands using the radioactive method, the ligands are first radioactively labeled, usually with iodide. Then, after determining the specific radioactivity of the ligands present, they are incubated with cells for a specified time. Some sort of separation (e.g. vacuum filtration) is performed to remove all unbound ligands, and the remaining radioactivity determines the amount of labeled ligands bound to receptors. Fluorescently labeled ligands are distinguished using spectrofluorometric methods. This method works quicker than the radioactive method because there is no removal step necessary. However, using fluorescently labeled ligands can cause cell damage, which is a negative. Two other less-often used methods of tracking ligands are electron dense markers and nanovid microscopy.

Experiments performed while tracking ligands can be executed using whole cells, cell membranes, or just isolated receptors. Each different method will produce slightly different results for the ligand binding rates. While using whole cells is the most realistic experimentally, the rates aren’t always constant due to the largest amount of trafficking present. Experiments using isolated receptors will often produce skewed numbers because they lack the other normal characteristics of the cell which are evident in every day life binding. The best method of experimenting with ligand/receptor complexes is to use either whole cells or cell membranes and account for the higher total binding rates.

Ligand/Receptor Binding
Non-specific binding occurs when the ligand is affected by membrane molecules other than receptors, or also when the ligands become trapped in the medium of the reaction. The rate of non-specific binding can be determined by measuring the rate of labeled “ligand binding” that occurs when there is essentially no receptors on which to bind. This rate needs to then be subtracted from the total ligand binding rate to determine the specific binding rate, or the actual ligand/receptor combinations which will send a signal.

The binding between ligands and receptors can be modeled using the reaction R	=	Free receptor number [#/cell] L	=	Free ligand concentration [$$M$$] C	=	Receptor/ligand complex number [#/cell] kf	=	Rate association constant [$$M^{-1}time^{-1}$$] kr	=	Rate dissociation constant [$$M^{-1}$$]

When this reaction is assumed to be at equilibrium, the equilibrium dissociation constant $$K_D = k_r/k_f$$ is used for analysis. A low value of $$K_D$$ indicates that the receptor is very willing to bind with the ligand. A high value of $$K_D$$ indicates the opposite, that they do not readily bind. As mentioned earlier, performing these experiments at a low temperature or a pH above 7 will decrease the trafficking effects. Although these are general trends found in experimental data, detailed information is not known on the exact effects of changing the temperature and pH.

Receptor Mobility
Along with trafficking, receptor mobility - commonly referred to as diffusion - also affects the ligand/receptor binding. Two techniques are used in determining these affects, fluorescence recovery after photobleaching (FRAP) and post-electrophoresis (PER). In both methods, the receptors are first fluorescently labeled. In FRAP, a small circular area of receptors is then irreversibly photobleached (a pulse of high intensity light is used to remove their fluorescense). The fluorescence of the area of the photobleached receptors is then continuously measured to determine the diffusion coefficient of the non-photobleached receptors. In PER, after the receptors have been fluorescently labeled, they are placed in an electric field so they align at a pole, and then removed from that field. Taking measurements of the receptors redistributing in the cell is then used to determine the diffusion coefficient. The FRAP method measures the self-diffusion coefficient, $$D_{self}$$, while the PER method measures the mutual diffusion coefficient, $$D_{mutual}$$. It has been determined that $$D_{self}$$ ≤ $$D_{mutual}$$ and also that they are equal to one another in infinitely dilute solutions of receptors. They will be assumed to be equal for the remainder of this article. Experiments usually will produce D values lower than the theoretical D values. This is attributed to receptor interaction with other molecules either present outside of the cell or on the membrane of that molecule.

Modeling of Receptor/Ligand Binding
One method that we can use to gain understanding about biosensors and how they behave is by modeling them mathematically. As in many instances, real world experiments are characterized by many variables that cannot be controlled or isolated. Modeling allows us to simplify the system to a point where only a few variables exist and may be studied. In this article a general model will be developed and then specific constraints and conditions will be applied to it.

Specific Single Receptor/Ligand Binding
To begin the analysis of receptor/ligand binding, we first examine the reaction between a single type of ligand and its corresponding receptor, which can bind to only one ligand at a time. The ligand (L) diffuses to the cell surface where it reacts with the receptor (R) to form a ligand-receptor complex (C). This reaction is reversible however, and the receptor may release the ligand. Thus we can represent this reaction as:  Assuming that these reactions are elementary, and that the solution is well mixed, the rate expressions for this reaction are given as:  where $$k_f$$ and $$k_r$$ are the formation and dissociation rate constants respectively. Therefore, we can say that the change in bound ligand concentration is given by the rate of formation minus the rate of dissociation. $$\qquad (3)$$ In order to solve this differential equation, we must find the concentration of the receptor and ligand as a function of time.

At this time it is appropriate to discuss the units of the quantities involved. For the purpose of this article the concentration of the ligand will be given in terms of moles/volumes such as molarity (M). The receptor and ligand-receptor complex however, will be measured in #/cell and the cell concentration (n) is in cells/ volume. Thus, in order to convert the receptor concentration into more conventional units we multiply it by the ratio of n over Avagadro’s number: $$\qquad (4)$$

First let’s examine the concentration of the receptor, R. At any given time, the total number of receptors per cell, $$R_T$$, must be conserved. This number is given by the number of free receptors plus the number of bound receptors. A similar analysis can be carried out for the total/initial ligand concentration, $$L_o$$. These two equations can be plugged into the original equation to get an equation that is dependent on time alone. $$R_T= [R] + [C]$$$$\qquad (5)$$ $$\qquad (6)$$ $$\qquad (7)$$

Case 1: Constant Ligand Concentration
If it is assumed that the initial concentration of ligand is extremely high, then the binding of ligands will not significantly affect the concentration of free ligands. But how much is a high concentration? Looking at equation (7), it can be seen that if , then that entire term is approximately equal to the initial concentration. This greatly simplifies the differential equation. $$\qquad (8)$$ At equilibrium, the formation rate equals the dissociation rate and the above equation is equal to zero. The equation can be rearranged to solve for the concentration at equilibrium, which is given by: $$\qquad (9)$$<br\> $$K_D$$is the ratio of the dissociation rate constant to the formation rate constant and is a measure of how strong the affinity between the receptor and ligand is. A small $$K_D$$ signifies a strong affinity (low dissociation/high formation) and a large $$K_D$$ signifies a weak affinity (high dissociation/ low formation).

The differential equation (8) can be solved analytically by integration to give a transient solution if an initial condition is given. We will assume that the generic initial concentration of bound complexes is $$C_0$$.<br\> $$\qquad (10)$$<br\> Equation (8) could also be analyzed using numerical methods such as Euler’s Method or the Runge-Kutta Method or by using software such as Polymath®. Below is a graph from such a method. It can be seen that there are two distinct sections of the graph. Before t = 50, the solution is transient and can be modeled by (10). After t = 50 the solution is near equilibrium and approaches the value given by (9). <br\> <br\>

This system may also be analyzed using dimensionless numbers such as u = [C]/$$R_T$$ and τ = $$k_r$$t. When u is plotted versus τ, constant values of $$L_o/K_D$$ give similar solutions. Plots of u versus τ with several values of $$L_o$$/$$K_D$$ can be made so that if $$L_o$$, $$K_D$$, $$k_r$$, and $$R_T$$ are known, [C] can found at any time by using the correct $$L_o$$/$$K_D$$ line. Rearrangement of (8) gives the differential equation which can be manipulated in the same way that was to get $$u_{eq}$$ and u(τ). For explicit demonstration, the reader is encouraged to reference the cited work. Also see Example 3 for an application of the dimensionless numbers.

Case 2: Ligand Concentration Depletion
When the initial ligand concentration is not large enough to assume that [L] does not change, then the solution becomes more complicated. Therefore dimensionless numbers are used again; however another quantity, η = (n$$R_T$$)/(Av$$L_o$$), is needed to describe the system. These are tabulated by making a plot of u versus τ for a single value of $$L_o$$/$$K_D$$ and several lines of constant η. The reader is once again refered to the cited works for explicit examples.

Case 3: Non-specific Binding
The above examples assumed that the effects of non-specific binding were subtracted  from the data. This section now builds the effects of non-specific binding into the model. The non-specific binding into the product B follows a similar reversible reaction that can be modeled by the rate equation<br\> $$\qquad (11)$$<br\> where $$k_{fn}$$ and $$k_{rn}$$ are the formation and dissociation reaction rate constants for non-specific binding respectively. The total ligand concentration is once again used to define L, but this time [B] and [C] must be used.<br\> $$\qquad (12)$$<br\> It is assumed that the rates of non-specific binding are much quicker than specific binding so non-specific binding is assumed to be at equilibrium, setting (11) to zero. Solving this equation gives<br\> $$[B] = K_N[L]$$$$\qquad (13)$$<br\> where and has a similar meaning as $$K_D$$. By substituting (13) into (12) and subsequently substituting (12) into (7), the change in [C] with respect to time may be modeled.

Determination of Parameters
There are two methods that can be used to determine the specific binding parameters, $$R_T$$ and $$K_D$$, assuming no non-specific binding. They are graphical methods and numerical methods. By rearranging (9), it can be seen that a linear relationship exists between $$C_{eq}/L$$ and $$C_{eq}$$.<br\> $$\qquad (14)$$<br\> $$K_D$$ can be found from the slope and $$R_T$$ can be found from the y-intercept. This plotting method is known as a Scatchard plot. Numerical methods such as nonlinear regression can also be used to try and fit these two parameters to the data. This is needed when the modeling equations are too complex and cannot be fit to a linear relationship.

Kinetic parameters can be found in a similar way for a transient solution with negligible ligand depletion where the following equation applies.<br\> $$\qquad (15)$$<br\> Since $$K_D$$ is known from the above method and $$L_o$$ is specified in the experiment, the slope of $$ln[C_{eq}-C(t)]$$ versus t can be used to find $$k_r$$. The parameter $$k_f$$ can be found then by dividing $$k_r$$ by $$K_D$$. These two parameters may also be found without going to equilibrium by choosing initial parameters that simplify equation (10). They are known as association experiments ($$L_o$$>0 and $$C_o=0$$) and dissociation experiments ($$C_o$$>0 and $$L_o=0$$).

Worked out Example 2
Frank the Tank, one of your buddy biologists, has asked you for some help in determining the ligand/receptor reaction properties for TNF, a common ligand family. Given the following data acquired during the experiments of different $$L_0$$ and $$C_{eq}$$ values, determine $$R_T$$ and $$K_D$$ by: a)	Numerical methods''' b)	Graphical methods, using a Scatchard Plot c)	Then, knowing $$k_r$$ = 0.14 $$min^{-1}$$ at room temperature, construct a plot showing how the ligand/receptor complex concentration changes with respect to time during the experiment with $$L_o$$ = 6.89E-09 M. Assume: 	The initial concentration of ligand/receptor complexes = 0 	        Non-specific binding has already been accounted for and 	        A constant ligand concentration is present

Solution: a) Non-linear regression is carried out using equation (9) as a model. This can be done by using software such as Excel, Mathematica, or Polymath.  For a tutorial on using Excel for nonlinear regression go to Excel Modeling    (9) The following is a view of the regression done using Excel: As you can see in the Excel regression, the value for $$R_T$$ is calculated as 6636 $$receptors/cell$$ and the value of $$K_D$$ is calculated as 1.532E-10 M. b) To obtain a solution using a Scatchard Plot, the above equation will need to be manipulated into a linear format such that the dependent and independent variables can be determined. This linearization is shown below (14) Then, using the values given, construct a plot of $$C_{eq}/L_0$$ vs. $$C_{eq}$$. The graph below is what was produced using Microsoft Excel. Knowing that $$-6.68E9 = -1/K_D$$, you can calculate $$K_D$$ to be 1.50E-10 M. Then, knowing $$K_D$$, $$R_T$$ can be calculated knowing $$4.41E13 = R_T/K_D$$, $$R_T$$  = 6615 receptor/cell. c) Using equation (8),, the concentration as a function of time can be found using numerical methods such a Euler's Method or the Runge-Kutta Method. $$k_f$$  can be calculated from  which gives the value of 9.3E8 $$M^{-1} min^{-1}$$.  The following graph was produced using Euler's Method with a time step of 0.01 minutes, showing the concentration of complexes vs. time.  For more information on the use of numerical ODE modeling in Excel, see Excel Modeling ODE Also the attached file "Biosensor Example 2.xls" may be used to see the worked out example. [[Media:Biosensor Example 2.xls]]

Worked out Example 3
You are a secret agent known as Agent "McCabe" who works for the US government. One day you and your partner, Agent "Thiele", are out doing secret agent things when out of no where a paper airplane lands in front of you. You pick it up hesitantly, unfold it, and notice that it is a note. It reads:<br\> "Agents McCabe and Thiele,<br\> Intel has just found out that the bad guys have just developed a new biological weapon. It contains a new ligand, ligand X, which binds specifically to the insulin receptor.  Only one ligand X can bind at a time.  We were able to find out that the total receptor concentration, $$R_T$$, is 1.0E5 receptor/cell, the $$K_D$$ is 4.2E-8M, and the $$k_r$$ is 0.4 $$min^{-1}$$.  If the initial ligand concentration is 8.4E-8M, how long will it take for the bound complex concentration to become fatal?  The fatal concentration is 5.9E4 complexes/cell.  You may assume that there is no non-specific binding and that the ligand concentration is constant.  This message will now self-destruct."

You know that there is some differential equation that could be used to solve this, but you have forgotten it. Therefore you pull out your field manual and you start flipping through the pages. Suddenly you notice the following chart:<br\> You think to yourself, "What luck!!! Or perhaps this was fate." Either way you continue to find out the time to reach the fatal concentration.

Solution:<br\> First you find the $$L_o/K_D$$ ratio.<br\> $$L_o/K_D = (8.4E-8M)/(4.2E-8M) = 2$$<br\> Thus we are looking at only the $$L_o/K_D = 2$$ line. Next we compute u.<br\> $$u = C/R_T = 5.9E4/1.0E5 = 0.59$$<br\> Looking at the graph, the τ value of the $$L_o/K_D = 2$$ line at u = 0.59 is about 0.7. From this we can compute the time.<br\> t = τ/$$k_r = 0.7/0.4 min^{-1} = 1.75 min$$