*Step 1: formulate the model*

The first step is to determine the state variable of the model. Given the above description, what is the most natural quantity to describe what quantity about the embryo is changing from day to day? The

of the embryo

Choose a variable to represent this quantity:

. Choose a variable for time (either `n` or `t`):

.

With your choice of variables, $＿_{ ＿ }$ represents the

of the embryo that are present

days after conception.

After fertilization (in day 0), the embryo consists of a single cell. (OK, it should be called a zygote until if divides into multiple cells, but you get the idea.) Therefor the initial condition is $＿_0=$

Write the dynamical system both in difference form as

$＿_{ ＿+1} - ＿_{ ＿ } =$

$＿_0=$

and in function iteration form as
$＿_{ ＿+1} =$

$＿_0=$

The function iteration result is easy to understand because the number of cells in one day is

times the number from the previous day. To understand the difference form result, you need to deduce that for any given day, the *change* in the number of cells from the previous day to that day is equal to

times the number from the previous day.

##### Hint

One tricky part of setting up the dynamical system is deciding whether it is easier to start with the difference form (such as $x_{t+1}-x_t =$ something) or the function iteration form (such as $x_{t+1}=$ something). If you choose to start with the difference form, then on the right hand side, you need to put the change in the number of cells. (And, if the number of cells is doubling, the *change* is **not** twice the previous number of cells.) If you choose to start with the function iteration form, then on the right hand side, you need to put an expression that gives the total number of cells after a day. (And, if the number of cells is doubling, the total number is indeed twice the previous number of cells.)

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*Step 2: solve the model*

Solve the dynamical system to obtain an expression for the number
of cells as a function of the number of days since fertilization.

$_＿^{＿} = $

##### Hint

Since the increase in the number of cells is proportional to the number of the previous day (i.e., we have a linear dynamical system), we will get exponential growth. In each day, we multiply the number of cells by what number?

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*Step 3: predict embryo growth and evaluate the merits of the model*

After one week, how many cells would there be in the embryo?

After two weeks?

After three weeks?

These values seem

, so we might

that our model is capturing the dynamics of the embryo growth during these first weeks.

What about the predictions of the model for the later stages of the pregnancy? If a pregnancy lasts 40 weeks, what does the model predict for the number of cells that the baby would have upon being born?

How does this number compare to the number of atoms in the
observable universe, which is estimated to be about $10^{80}$? The number is about

times

than the number of atoms in the observable universe.

Do you believe the results of the model for the end of the pregnancy?

What does the model assume about the rate of cell division? It assumes that the rate of division

throughout the pregnancy. That assumption must be

.

What must be true of the rate of cell division? Can the cells continue to divide into two each day for the whole course of the pregnancy?

##### Hint

Since you solved the system in the previous step, you just need to plug in the number of days of a pregnancy into the solution formula.

If, for the end of the pregnancy, the number of cells is ridiculously large, then what can you conclude about the rate of cell division? It is at all possible that this model is correct, i.e., that the rate of cell division continues at a constant rate of one division per day for the whole pregnancy? If the model is clearly bad, then what must be happening to the rate of cell division?

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