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Mathematical Modelling and Computer Simulations in Undergraduate Biology Education.

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Spreadsheets in Education (eJSiE)
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Mathematical Modelling and Computer
Simulations in Undergraduate Biology Education
Gavin A. Buxton
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Mathematical Modelling and Computer Simulations in Undergraduate
Biology Education
Abstract
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Keywords
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1 Introduction
Undergraduate biology education can be enhanced through the introduction of mathematical mod-
els and computer simulations by improving general quantitative skills, ensuring students are more
comfortable interpreting the results from computational studies and encouraging students to im-
plement computer models as part of their future research. There have been many studies that have
linked the use of computational tools in STEM (science, technology, engineering and mathematics)
classrooms to the enhanced learning of the students [1]. This is potentially more beneficial in bi-
ology classrooms as, in contrast to other STEM disciplines, biology majors are disproportionately
composed of more female students [2]. Gender identity within our society often reinforces a per-
ception that boys prefer computational, engineering or mathematical based careers while girls show
a preference for the life sciences [3]. A potential gender gap in the use of mathematical models
and computer simulations in future biomedical research could be alleviated through the use of a
computerized learning environment, and by increasing the familiarity and proficiency of biology
undergraduates with computer simulations in female-dominated classrooms [4]. In addition, there
is strongly believed to be a technological gap between working and educated classes, the former in-
cluding disproportionately many African Americans and other people of color, and bridging this gap
at the university level is vitally important for promoting the diversity of future biology researchers
[5, 6]. Given the important role that computer simulations will play in biological research (research
in areas such as sustainable food growth, sustaining ecosystems and biodiversity, and understanding
human health [7]) it is important that all of our students (regardless of race, gender or socioeco-
nomics) are proficient in the use of mathematical models and their exploration with computational
simulations, and better prepared to meet these future challenges [8]. This is in contrast to how
mathematics, computer science and biology courses are currently taught. Mathematics is taught
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almost entirely independent of undergraduate biology courses [9, 10], with students unable to ap-
preciate the influences and overlap between these disciplines. Even more alarming is that biology
majors often take few or no computer sciences classes. This has a direct and measurable impact on
future biologists and their research, with papers containing a greater concentration of mathematical
equations often receiving less citations by their peers [11, 10]. This lack of comfort that biologists
have towards mathematically-dense papers limits their ability to appreciate the increasingly larger
role that computer simulations are playing in biological research [10].
The majority of undergraduate biology textbooks are densely filled with facts and images, and
lack insights into the nature of scientific inquiry [12] and the computational skills required for future
biologists [10, 13, 14]. This is changing, however, with computer simulations increasingly being
used in undergraduate biology courses [15]. It has been argued that computer simulations can allow
students to perform experiments that might be too impractical, dangerous or expensive in reality [16,
17]. Furthermore, at the high school level there are concerns for animal welfare, and arguments have
been made for replacing traditional dissections with computer simulations [17]. This often results in
computer simulations being used as a black box tool for students to interact with on the short-term
[17]. Some of the computational tools that are used for enhancing instruction, however, are both
complicated and enlightening; for example, the Quantitative Circulatory Physiology model exposes
future healthcare professionals to many physiological scenarios that their patients might experience,
including the simulation of heart failure, anemia, and diabetes [18]. On the other hand, computer
simulations are not simply an alternative tool to traditional instruction. Computer simulations
can conceptually elucidate and quantitatively explore the fundamental science and mechanisms
that comprise enormously complex biomedical systems, and this is making mathematical modeling
and computer simulations an increasingly integral part of biological research [14]. Mathematical
modeling and computer simulations, therefore, should be taught to undergraduate biology students
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in an interactive manner, which allows students to directly implement and change the fundamental
mathematics and computational implementation of these simulations. In this manner, educational
computer simulations become not only a set of evolving pedagogical tools [14] but should reflect
the inquisitive and investigative nature of computer simulations in biological research. Introducing
computer programing to undergraduate biology students is not only important for establishing a
future generation of computational biologists, however, but also improves students use of scientific
logic and comfort with quantitative analysis [19], which are required skills for all future biologists
[10].
Spreadsheets are important tools for biologists and an excellent tool for introducing biology stu-
dents to mathematical modeling and computer simulations. Spreadsheets are a particularly useful
teaching resource at the university level because essentially all classrooms in U.S. public schools
(and classrooms in most developed countries) have access to computers and the internet [20, 21].
Futhermore, spreadsheets are increasingly being incorporated in to online environments (such as
course management systems and ebooks). That said, biology students can still often struggle with
using spreadsheet software and it is important that students recognize the difference between the
science they are trying to elucidate and the computational mechanisms of the spreadsheet envi-
ronment [19]. However, the widespread use of spreadsheets both in biological research, and in the
workplace, make spreadsheets the obvious computational environment to introduce computer pro-
graming to undergraduate biology students. Spreadsheet models can also be very complex [22]. For
example, Geyer recently implemented the Hodgkin-Huxley Model for action potentials in neurons
using a spreadsheet model, allowing the students to explore this model and design their own exper-
iments to further their understanding of neuron behaviour [23]. As another example, Langendorf
and Strode recently implemented a simulation of an evolving population experiencing natural selec-
tion to introduce evolution in the classroom [24]. The inherent complexity can sometimes make the
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behaviour of these systems difficult for both students and researchers to intuitively deduce, and the
use of computer simulations can be invaluable to elucidating the nature of these systems. Further-
more, mechanisms within a computer model can be systematically (and sometimes unrealistically)
turned off in a simulation to explore the strength and behaviour of different effects and interactions.
Here, we describe a computational biology course that introduces computer simulations to un-
dergraduate biology students using a spreadsheet environment. The course both introduces the
students to computer models, and also incorporates a research component that sees students im-
plement computer models found within the scientific literature and adapt these computer models
as part of an in-class scientific research project. Furthermore, a couple of examples of computer
simulations that are implemented by our students as part of the course are detailed. The spread-
sheet simulations described here are the Lotka-Volterra predator-prey model, a cellular automaton
model of tumor growth, and a model of an infectious disease outbreak.
2 Computational Biology
Computational Biology is a course taught at our institution that uses spreadsheets to introduce
mathematical modelling and computer simulations to our undergraduate biology students. Students
investigate models that explore a variety of different biological systems. The prerequisites for this
class is General Physics I (a calculus-based mechanics class) which our students will usually take
in their first or second years at university. Students are, therefore, familiar with the concept of
mathematical models of the physical world and will already have taken Calculus I, which covers
differentiation and introduces integration. No prior programming experience is required before
students take this class, which makes spreadsheets the perfect programming environment for this
course. Students also work on a research project and learn how to present their results in a scientific
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manner.
The first part of the course introduces students to spreadsheets. In particular, we describe
the basics of entering data, entering formula and some of the functions available to spreadsheets.
Furthermore, we discuss the inclusion of macros and the addition of a button within the spreadsheet
that runs the macro; whilst spreadsheet syntax is generally universal, it should be noted that macros
(or scripts) are handled differently for different spreadsheet software.
In terms of mathematical formula, our students have not been introduced to finite difference
calculations and numerical integration before taking this course. Therefore, the course covers the
derivation of forward, backward and central difference approximations and the trapezoidal rule. It
is important for students to understand the role that discretization has when numerically simulating
a continuum mathematical model. In particular, the role of the temporal and spatial discretizations
in the numerical stability of the computer simulation is an area that should be emphasized.
As an example we first consider diffusion in one-dimension. The diffusion equation is given by
∂φ
∂t =
∂x D∂φ
∂x (1)
where φis the concentration, tis time, Dis the diffusion coefficient, and xis position. The discrete
form of the equation that is implemented into the spreadsheet is of the form
φt+1
i=φt
i+Dt
(∆x)2φt
i+1 +φt
i12φt
i(2)
where the superscript represents the discrete time, and the subscript represents the discrete location.
This gives students a way to explore numerical stability and understand how the discretization in
time and space, ∆tand ∆xrespectively, influence numerical stability.
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Another simple example that we initially consider is the logistic growth model, which captures
the growth of a population that is constrained by resources or its environment. The rate equation
is of the form
dN
dt =rN KN
K(3)
where Nis the population, ris the growth rate, Kis the carrying capacity and tis time. The
discrete form of this equation is
Nt+1 =Nt+ ∆trN tKNt/K (4)
where the superscript represents the discrete time. This is a good example of growth rate that
students may have seen in previous courses (without really looking too closely at the mathematics,
and certainly without numerically implementing this model in a spreadsheet).
The course also introduces students to research methods and the process of writing scientific
papers. Students would typically take this class prior to starting their senior thesis research project.
The course then progresses by alternating each week between students working on their in-class
research project and students implementing a model that is chosen by the instructor. The following
sections give examples of some of the models that students implement in this course.
3 Lotka-Volterra predator-prey model
The Lotka-Volterra Predator-Prey System is usually captured with the following nonlinear differ-
ential equations that can result in a continual cycle of growth and decline [25]. The rate of change
of the number of prey is given by
dx
dt =αx βxy (5)
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where xis the number of prey, yis the number of predators, αis the prey growth rate and βis the
rate of predation. The rate of change of the number of predators is given by
dy
dt =δxy γy (6)
where δis the predator growth rate and γis the predator loss rate. These simple coupled equations
can lead to quite interesting behaviour.
These equations can be discretized and implemented in a spreadsheet environment by the stu-
dents to explore the complex interactions between predators and prey in this model. The discretized
equations are simply
xt+1 =xt+ ∆tαxtβxtyt(7)
and
yt+1 =yt+ ∆tδxtytγyt(8)
where ∆tis the time step and superscripts represent discrete time. An implementation of this model
is depicted in Figure 1. This is a wonderful example of the dynamics of a model which undergoes
cyclic behaviour. For undergraduate biology students, whose only experience of calculus is a course
that covers very basic differentiation (always with respect to one variable) and a brief introduction to
integration, implementing these coupled equations and observing the resultant dynamic behaviour,
can be quite thought-provoking. Once students have implemented this model they can try to see if
they can capture the dynamics of a real system; real systems invariably exhibit greater complexity
than the models we use to mimic their behaviour and this can result in a good area of discussion. For
example, some of my students had already taken a zoology course with a colleague whose research
interests is in amphibians, and this resulted in a classroom discussion about the applicability of
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Figure 1: Example of the spreadsheet implementation of the Lotka-Volterra predator prey model.
The system is cyclic. As the predators eat the prey their population increases, but the number of
prey decreases. This is unsustainable and the number of predators will decrease once the number
of prey is sufficiently low. However, as the number of predators decreases the number of prey is
allowed to rebound, and with it the number of predators, and the cycle reiterates once more.
this model to wood frogs. Wood frogs are found in Western Pennsylvania and have an interesting
ability to tolerate freezing temperatures. This allows the frogs to stay nearer the surface during
winter and emerge sooner in the spring and capture the prey first, before other frogs that have to
bury themselves further underground during the winter. The class discussion centered around how
we can mimic the effects of temperature in the model. The students were interested in the idea
that the properties of the model could be time-dependent and the “activity” of the freeze-tolerant
frogs and various insects could be slowly turned on to mimic the beginning of spring.
4 Cellular automaton model of tumor growth
This two-dimensional model of cancer growth is adopted from Poleszczuk and Enderling [26] and
can easily be implemented, on a smaller scale, within a spreadsheet model. Within this spreadsheet
model, each cell of the spreadsheet represents an individual cancer cell that occupies a spatial area
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of (10 µm)2on a two-dimensional regular square lattice.
Each cancer cell is characterized by its remaining proliferation potential (how many times left
that a cell can divide), ρ, and its probability of spontaneous death α. The model assumes a
heterogeneous tumor population consisting of cancer stem cells and non-stem cancer cells. Cancer
stem cells are assumed to be immortal and have unlimited proliferation potential. In other words,
their remaining proliferation potential is infinite, and their probability of spontaneous death is
zero. Non-stem cancer cells, on the other hand, can only divide a limited number of times, ρmax,
before cell death. Each cell type can divide symmetrically to produce two daughter cells with
parental phenotype. In other words, a stem cell would proliferate two stem cells, but a non-stem
cell would proliferate two non-stem cells (with a remaining proliferation potential reduced by 1).
However, a cancer stem cell can also undergo asymmetric division and create a cancer stem cell
and a non-stem cancer cell with ρ=ρmax (and the remaining proliferation potential of these non-
stem cells would decrease with each subsequent non-stem cell division like any other non-stem cell).
The probability of asymmetric division is 1 psy mm (where psymm is the probability of symmetric
cancer stem cell division). Cells need adjacent space for proliferation, and cells that are completely
surrounded by other cells (the surrounding eight spreadsheet cells on our two-dimensional lattice)
become quiescent. Otherwise, cells can randomly proliferate into vacant adjacent space. Cells can
undergo spontaneous death, independent of the available space, with rate of α. Cells that die are
instantaneously removed from the system, and this space is then considered empty. The temporal
discretization in the model is ∆t= 1/24 day (i.e., 1 hour), and proliferation probabilities are scaled
to this simulation time step.
A layout of the simulation is depicted in Figure 2. The simulation (which consists of 20 ×20
cells in the spreadsheet) is replicated 14 times through the spreadsheet. The first instance is shown
as the left-most square in Figure 2 and contains the current state of the system. This is comprised
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Figure 2: Example of the spreadsheet implementation of a two-dimensional cellular automata model
of cancer growth. This is how the different stages of the variables during a single iteration are stored
across the spreadsheet. Starting on the left is the current state of the system, and moving across
to the right we end at the next iterative state.
of a number which represents the current remaining proliferation potential (or -1 if the spreadsheet
cell contains a cancer stem cell). Immediately to the right of this is the next 20 ×20 grid that
contains either 0 if the cell is going to spontaneously die or maintains the remaining proliferation
potential if the cell survives onto the proliferation stage of the simulations iteration. For any cells
which are non-stem cancer cells we check to see if a random number is less than the probability of
the cell dying (in other words, if the random number is less than αthen the cell is removed).
Next, we have to capture the proliferation of the cancer cells. If a random number is less than
the probability of proliferation and the space is occupied than the cell is identified to proliferate
(the value within the spreadsheet cell is set to 1). Below this is another two 20 ×20 grids which, if
the cell is identified to proliferate, is assigned a value of between 1 and 8 that represent directions
that the cancer cell will attempt to proliferate in.
In figure 2 there are now a series of eight 20 ×20 grids arranged in the directions that the
proliferation can be in. We go from one direction to the next and if the cell is empty, and the cell in
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Figure 3: Example of the spreadsheet implementation of a two-dimensional cellular automata model
of cancer growth. a) A screenshot of the spreadsheet, showing the variables at the top and the
current state of the simulation below, and b) a contour plot of the remaining proliferation potential
(or -1 if the spreadsheet cell contains a cancer stem cell).
the opposite direction is proliferating in this direction then it will contain a new cell. The identity
of the new cell is randomly determined to be either symmetrically proliferating or asymmetrically
proliferating (creating both stem and non-stem cells) based on the probability psymm. The new
remaining proliferation potential is determined (or the spreadsheet cell contains -1 if the cancer
stem cell proliferates another cancer stem cell). In other words, if the cell is occupied by a stem
cancer cell then the new cell can either by a stem cell (equal to -1) or a non-stem cell with a
proliferation potential set to the maximum value, but if it is occupied by a non-stem cell then the
proliferation potential of the newly created non-stem cell is reduced by 1. Note the probability of
the stem cancer cell proliferating either a stem cell or a non-stem cell depends on the probability
of symmetric proliferation This is calculated for each direction in turn and then copied to the last
20 ×20 grid, on the right of Figure 2, which represents the next iteration of the simulation.
Now to complete an iteration we just have to copy the next iteration grid over to the original
grid. We can create a macro that will simply copy the contents and paste special (the values but
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not the formulas) to the original cells, and insert a button that when pressed will run the macro to
update the system. Figure 3a shows a snapshot of the spreadsheet simulation. The constants are
stored at the top of the spreadsheet and the current state of the system is below it (the top left
20 ×20 grid from Fig. 2). At the top of the screenshot is the button that will run a macro that will
copy the next iteration values back to the current iteration values. Once these values are copied the
spreadsheet will automatically recalculate all of the values for the next iteration. In other words,
every time the button is pressed the simulation progresses through one iteration (representing an
hour of time). Figure 2b shows the remaining proliferation potential (or -1 if the spreadsheet cell
contains a cancer stem cell) after a number of iterations. For the values considered here there would
appear to be a greater concentration of stem cancer cells in the center of the tumor.
During the class discussions that have followed students seem skeptical and believe that the
model might not necessarily represent a real tumor. In particular, the two-dimensional array of
numbers in the spreadsheet can be difficult for students to visualize as an actual representation of
a tumor. Discussions in this class can sometimes be a little uncomfortable as there will be students
who have lost family or friends to cancer. I generally steer the conversation to cancer treatments.
For example, using nanoparticles to deliver chemotherapeutic agents [27], immunotherapy [28] or
molecularly targeted therapy [29]. The model predicts that the stem cells are generally contained
in the center of the tumor, and this has lead the discussion on to the accessibility of different parts
of the tumor to nutrients (and the administered drugs) and the emergence of a complex vasculature
in tumors. Interestingly this can result in an enhanced permeability and retention (EPR) effect
which sees small particles tending to accumulate in tumor tissue to a much greater extent than
they do in normal tissues. Furthermore, there is a strong debate on the nature of the emergence of
drug resistance during chemotherapy that is worth mentioning. The debate surrounds whether drug
resistance is initially present in the tumor to some small extent, or if the drug resistance emerges
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while the tumor is exposed to treatments than can increase mutation rates. In the latter case,
does having stem cells in the center of a tumor surrounded by non-stem cancer cells that are being
exposed to chemotherapeutic agents result in a greater probability of drug resistance emerging?
5 Model of an infection outbreak
The models that represent the outbreak of an infection are well-established and this computer
simulation is used to introduce these models to the students. We choose an infection that the
students are familiar with for this example, and simulate the spread of a zombie apocalypse [30].
This lecture coincides with the week of Halloween and is a fun way for the students to explore this
class of model. Furthermore, students often get excited about which zombie movie or television
show is their favourite, and attempt to vary the parameters to best replicate the effects of their
chosen strain of zombie outbreak.
The differential equations for this model are
dS
dt = Π βSZ δS
dI
dt =βSZ ρI δI
dZ
dt =ρI αSZ
and
dR
dt =δS +δI +αSZ
where Sis the number of susceptible people, Iis the number of people infected, Zis the number
of zombies and Ris the number of people permanently removed from the system (dead, but not
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undead). Π is the birth rate of new people, βis the transmission factor (how people become
infected), δis the rate at which people die from non-zombie related causes, ρis the rate at which
the infected become actual flesh-eating zombies, and αis the rate at which humans kill the zombies.
We will solve these equations using the usual finite difference equations (Euler method) and
choose our time step, ∆t, to be suitably small. The discretized equations, that are suitable for
incorporating into the spreadsheet, are
St+1 =St+ ∆tβStZtδSt]
It+1 =It+ ∆t[βStZtρItδIt]
Zt+1 =Zt+ ∆t[ρItαStZt]
and
Rt+1 =Rt+ ∆t[δSt+δIt+αStZt]
where the subscripts represents the time. A screenshot of the spreadsheet simulation is included as
Figure 4. The various constants are placed at the top of the spreadsheet and directly below this are
5 columns. The first column is time, with each subsequent cell just being updated by the value of
t. The second column is the number of susceptible people and this is increased by the birthrate
(which is relatively small), significantly decreased by the transmission of the zombie “disease” and
slightly decreased by the rate at which people die from non-zombie related causes. The third column
is the number of people who are infected with the zombie disease (bitten by a zombie). This is
increased by the same rate of transmission of the zombie “disease” that significantly decreased
the number of susceptible people, but is also decreased by the rate at which the infected become
zombies and (similar to the susceptible population) slightly decreased by the rate at which people
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die from non-zombie related causes. The fourth column is the number of zombies. This is increased
as the infected turn into zombies and is only decreased by the rate at which susceptible people kill
the zombies (although students who reference the movie 28 days later argue that over large times
the zombies will die off on their own). The final column is the number of people removed from the
system. This is increased by the death of susceptible and infected people, or the death of zombies
at the hand of normal humans; note that in the current model this saturates at α/β, as with the
variables considered here the death of zombies is much more prominent than the regular death rate
(although we might expect this to increase during the apocalypse). The zombies quite quickly take
over the world in the current model.
This provides a wonderful way for students to interact with this model and understand the
relationship between the terms in the equations and the overall behaviour of the system. This
model also allows students the opportunity to add additional terms to the equations (or even
populations) to make the model more accurately capture the behaviours found in their favourite
zombie movies. As an example I generally start the discussion by comparing Night of the Living
Dead slow-moving zombies to the 28 Days Later fast-moving zombies. In the case of fast-moving
zombies we might expect the probability of zombies infecting humans to be much higher and the
probability of humans killing zombies to be much smaller. Furthermore, the George A. Romero
zombies can infect susceptible people and the infection leads to zombification over several hours,
while Alex Garland imagined the zombie transformation to be almost instantaneous. This can be
included by drastically increasing the rate that infected are transformed to zombies. The most
interesting discussions, in my opinion, relate to areas that are not initially part of the model. For
example, in one class a couple of students modified the model to account for susceptible people
killing infected people. I think this is a really interesting development. Not just taking a model
and changing the parameters, but exhibiting the confidence and creativity to modify the model to
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Figure 4: Example of the spreadsheet implementation of a model of an infection outbreak. The
infection is chosen to be a zombie apocalypse and the number of susceptible, infected, zombified,
and dead people are calculated and plotted as a function of time.
capture the behaviour they want to mimic.
6 Summary and conclusions
Here we have described a course which we offer that introduces our undergraduate biology students
to computational thinking, mathematical modeling, computer simulations and the power of spread-
sheets in implementing simple models. The course stresses the role of computer simulations in
scientific research, and the benefits of computer literacy and quantitative skills throughout biologi-
cal research. Three examples of models implemented in a spreadsheet environment by our students
have been detailed: the Lotka-Volterra predator-prey model, a cellular automaton model of tumor
growth, and a model of an infectious disease outbreak. Other computer simulations also currently
explored within this course include the dynamic instability of microtubules [31], the treatment of
drug-resistant strains of tumor cells [32], a tissue heat transfer model of scalds and burns [33], and
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a model of dry-eye syndrome [34]. However, the possibilities are endless and the choice of models
to explore within the class is largely based on the interests of the instructor and students.
We have found that the incorporation of computer simulations within undergraduate biology is
important for improving the quantitative skills of our students, and their ability to use spreadsheets
to analysis experimental results. This is perhaps not too surprising after an entire semester course
where the students implement computer models in a spreadsheet environment. Furthermore, a
number of students have subsequently gone on and incorporated spreadsheet models within their
senior thesis research projects and honors research projects (outside of this course). In particular, in
the last two sections offered (which contained on average 15 students each) 6 biology students have
continued using computer models and simulations in their future research projects; something that
would have been unheard of before this class was offered. The long term impact of this course is,
therefore, expected to be significant in the research abilities and possible research direction of our
graduating students. Certainly we hope that the observation that biologists are uncomfortable with
articles that contain mathematical models and computer simulations would not apply as much to
our students after taking this course. As computer simulations and computational thinking become
increasingly prevalent in K-12 education [35] we might expect that undergraduate biology students
will become increasingly comfortable with integrating quantitative skills into their biology research.
Socioeconomic factors will see this occur in some demographics sooner than others, and including a
comprehensive introduction to computer simulations within an undergraduate biology sequence is
expected to help diversify the make up of future computational biologists. Regardless of a student’s
background, however, I have observed that all students that have taken this course have developed
an ability to implement and manipulate computer models within a spreadsheet environment, and I
believe that their general spreadsheet skills have improved significantly.
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