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What is an example of a core concept and some variations?
Frank's research over the past 20 years has focused on reentrant
cardiac arrhythmias and the potential role antiarrhythmic drugs might
play in amplifying the potential for triggering reentrant arrhythmias.
Under normal condition, the membrane potential of a group of pacemaker cells
oscillates with a frequency of about 1/min. Each time the membrane potential
exceeds a threshold, neighboring cells are excited and a wave of excitation
propagates away from the pacemaker region. Because the heart is a closed
surface, this wave will eventially collide with itself and thereby is
extinguished (due to a property called refractoriness).
A reentrant arrhythmia is one where the excitation wave circulates around the
heart without colliding with another wave and therefore is capable of
reexciting the heart. Clearly, a continuous front can never become reentrant.
However a discontinuous front can evolve into a reentrant process [16].
The variations of the theme of discontinuous fronts
are all the different ways one can make a discontinuous wave: by
premature excitation, by collision of a front with an obstacle, by
excessive front curvature and by encountering nonuniform refractory states.
Each variation has
specific detail that is required for the mechanism to successfully
function within a specific environment.
Typically, one begins by creating a model of a specfic event or
phenomenon. However, over time, one might notice that the model
applies to other events or phenomena and can be used to answer
questions that are completely unrelated to the original intent, thus,
demonstrating the potential for generalizing the model.
To be able to demonstrate that a model, as representing some physical
mechanism, generalizes to describe processes in many different
settings is the greatest thrill possible.
The main challenge in biology is identifying processes, mechanisms
and developing an understanding of the minimally complex representation.
But before starting to model, we ask, about what features the model
must represent. For example, we view a living organism as requiring
6 essential processes: metabolism (converting nutrients to energy sources),
translation (translating an electrical signal to motion),
signaling (transfering the representation of an event from one place to
another), replication (duplicating something) and regulation,
If we model an organism,
then probably these features must be included in the model. Such models
are quite useful, because we can use the model of one entity as a template
for investigating and characterizing another. For example, at the level
of the nucleus. expression could be considered as replication, signaling
could reflect the initiators and terminators of expression, metabolism
could reflect the supply of raw materials to the expression system etc.
For all these complexities, though, it seems that
linear models are adequate to describe many processes. Not that these
processes are inherently linear. Rather most likely, the range over
which we can explore them is small, and the processes appears linear.
Figure:
A Saturating Process: A frequently occurring nonlinearity
in biological systems
A doseresponse curve of a saturable receptor reaction where
unoccupied sites and occupied sites.

In the case
of a finite number of receptors, with
a huge drug concentration, all receptors are occupied  and
increasing the concentration does nothing.
Similarly, at very small concentrations, no receptors are occupied  and
at intermediate concentrations, some receptors are occupied, leading to the
sigmoid shaped doseresponse curve:
represented by
where
b is the fraction of bound receptors, l is the
reverse rate constant, k is the forward rate constant and D is the
drug concentration. When D is small, b = 0 and when D is large, b = 1. Below
we'll derive this relationship exactly.
Thus (see below) estimating
parameters derived from linear models is an important statistical
tool. We'll derive a simple least squares procedure and hypothesis
testing concept that is readily generalized to nonlinear and categorical
data models.
In addition, the fact that many nonlinearities are derived from saturable
processes results in simple linear approximations of the saturable process:
3 lines  one for low concentrations, one for intermediate concentrations,
and one for near saturable concentrations. Unless you happen to be operating
near the knees of a saturable process, linear models work really well.
One of the main products of statistical theory is
that parametric procedures (which assume normally distributed variations)
usually
give the same answers as their nonparametric sisters. This is due to
the central limit theorem. The central limit theorem says that sums of
random variates are asymptotically normally distributed. My
numerical studies indicate that when you have 7 or 8 terms in the
series, then asymptotic normality rules the day. Thus, we never worry about
the underlying distribution (well almost never) because we are analyzing sums
of random variates (mean, variance etc) which are asymptotically normal
(or chi square for sums of squared normal variants).
Model building can be based on algebraic equations or
differential equations. When do we use which?
The main idea with differential equations can be best viewed by
comparing with algebraic equations. Solving algebraic equations
results in finding points that
satisfy the equations, while solving differential equations
results in finding functions that satisfy the equations.
Here we show how differential equations arise in ordinary problems
and how to solve a simple first order linear ODE (ordinary
differential equation). With these tools, you can run over any boulder.
Modeling is simply
translating a physical process into some equations that describe the
physical process. Thus, the idea is to get a mental image of the
process to be modeled, then using basic physical and chemical concepts,
write the ODEs or PDEs that describe the process.
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Frank Starmer
20040519