Cell Biology: Containment: vesicles, vacuoles, and lysosomes

Tutoring biology, the cell is fundamental.  Heading towards another weekend Biology 12 workshop, the biology tutor recalls Bruce Willis’s advice in Last Boy Scout: “Be prepared.”

A cell, like anyone, needs storage.  More than that, it needs storage for different purposes.  While you keep your food in the pantry, you store your clothes in the bedroom closet.  The cell faces similar storage challenges.

The three main types of storage vessels a cell uses are vesicles, vacuoles, and lysosomes.  Each has its own particular use and features:

Vesicles are often used for transport.   For example, they are used to store molecules or food arriving from outside the cell.  Vesicles are also used to hold partially completed molecules the cell is making as they are moved to different “work sites”. Then, when a molecule is to be secreted from the cell, it is shipped to the outer membrane in a vesicle, then released outside.

A vacuole is larger than a vesicle.  Vacuoles are meant for storage until use – or even permanent storage.  Water, sugars, and even pigments are stored in vacuoles.  In the case of pigment, it will remain in the vacuole for the life of the cell, giving the cell color. A water vacuole holds a large amount of water in order to “fill out” the cell, giving it the proper shape.  Plants derive their rigid shape partially from the water in their cells’ vacuoles.  Vacuoles can also hold toxic by-products until the cell gets around to destroying them.  Although both plant and animal cells contain vacuoles, plants use them more.

A lysosome is a special type of vesicle that stores digestive enzymes (see my previous post). A vesicle containing food will be fused with a lysosome for digestion to take place.

A cell can be a busy place. Any such place needs ample storage. Fortunately for the cell, it can make new vesicles, vacuoles, or lysosomes as needed:)

Source: Inquiry into Life, 11th Edition. Mader, Sylvia. New York: McGraw-Hill, 2006.

Jack or Oracle Tutoring by Jack and Diane, Campbell River, BC.

Biology: What is an enzyme?

Tutoring Biology 12, you mention enzymes often.  The biology tutor defines the term enzyme, having used it in previous posts.

Recalling elementary school, your teacher likely led holiday crafting.  For example, she might have handed out pages with snowmen traced on them for the kids to cut out and decorate.  Likely, the finished snowmen were affixed to the window or wall.

Back then, photocopying was still pretty new; not all schools had a photocopier.  If they did, they didn’t use them much; likely, photocopies were expensive.  They still are, by the way:  going “over-budget” on photocopies can be a real problem in bureaucracies.

Therefore, the teacher didn’t use a photocopier.  She traced each snowman by hand.  However, she didn’t do it free-hand; she used a pattern.  Likely, she drew one “good one” by hand on cardboard, then cut it out.  To produce a snowman sheet for a child, she laid the cardboard pattern on a sheet of blank paper and traced around.  Tracing around the cardboard was quick: likely, she could produce twenty-five snowmen sheets just as quickly that way as by going down the hall to the photocopier, running them off there, then bringing them back.  What’s more, she could afterwards put the cardboard pattern away amongst her other supplies.  Next year, it would be there, waiting to be put to use again.  From one careful snowman drawing done on cardboard in her early days, she could produce unlimited snowman sheets over her teaching career.

Let’s consider a cell in the human body.  Like the teacher, it needs to repeat the same job(s).  To a cell, a job is a chemical reaction.

Does the cell do the reaction “free-hand”?  No.  Like the teacher, it makes a physical pattern that can be reused as many times as needed.  That pattern, to a cell, is an enzyme.

Most enzymes are proteins.  Like the teacher’s snowman pattern, the enzyme’s shape defines which job it helps with.  To the teacher, a different art project would require a different cardboard pattern.  To the cell, each specific reaction requires its own specific enzyme.

If the teacher could not use patterns, preparing for the art project might take prohibitively long; similarly, the cell’s life processes can not occur quickly enough without enzymes.  Some poisons specifically target enzymes and shut them down.

The veteran teacher found her job easy, since she’d accumulated such a helpful toolkit over her years.  Similarly, a cell whose enzymes are all available and functional is likely very prosperous:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

English: punctuation: the dash

Tutoring English, you get asked how to make writing more interesting. The English tutor offers a way to add variety to your sentences:  occasional use of the dash.

The dash is not the most commonly used punctuation mark; however, some people use it frequently.  It can mark a sudden change in thought, which is the way many people really do think.  Consider the following examples:

I bought him some wine, afterwards recalling he is a recovering alcoholic – what was I thinking?

I bought my bed at Robinson’s Furniture during Saturday Morning Markdown – I don’t know if they still do that.

We discussed the exam afterwards – none of us felt confident.

Dashes can give the flavour of how people truly talk and think.  They can also be used to enclose an idea, rather like parentheses:

If we invite Dave – we’ve invited him every year so far – we’ll need a vegetarian dish.

While we’re in Berlin – we’ll be there for two weeks – I hope we can attend the opera.

I might overuse dashes.  If you don’t use them already, start slowly:  put one or two in an assignment, then wait for feedback.  If your teacher doesn’t complain, you can start experimenting.  If, on the other hand, they do complain, try to comply with their advice the next time you use a dash.  You needn’t use them often in order to benefit from the variety they can add to your writing:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: break-even point

Tutoring math, you realize that to most people, math’s main importance is its applications.  The tutor presents a business example.

Any small business owner knows the break-even point is when you’ve paid your expenses, but haven’t “made” any money.  At the break-even point, you haven’t lost or gained.

In math, you can notice the break-even point on a graph.  (For a refresher in graphing, see my post here.) Consider the following example:

The Chess Club, in order to fund its overseas tournaments, holds a bake sale each year.  The items at the sale cost $1.50 each.  To rent the hall, the club pays $240.  Find the break-even point.

Solution:

Let P=profit
Let n=number of baked goods sold

Then P=1.50n – 240

Notice, of course, that profit doesn’t mean income; rather, it means income less expenses.  The only expense we are considering is the hall rental; the baked goods, we can assume, are donated.

At the break-even point, P=0.

Let’s look at a graph that models the situation.  To make some points to plot, we choose a few values of n, then plug each into our equation P=1.50n-240 to get the corresponding values for P.  For instance, let’s find P when n=100:

P=1.50(100) – 240

P=150-240

P=-90

Therefore, when n=100, P=-90.  By repeating that process for various values of n we arrive at the following table:

n (number of baked goods sold) P (Profit)
0 -240
100 -90
200 60

Each (n,P) from the table means a point on the graph. For instance, when n=100, P=-90; therefore, (100,-90) will be on our line. We plot the points from our table as follows:

You can see the break-even point: it’s where our line cuts the horizontal axis. Note that, in this case, the x axis has been renamed the “n” axis; similarly, the y axis is called the “P” axis. In word problems, textbooks often rename the variables, calling them letters that stand for elements in the specific problem.

In our case, of course, the break-even point is 160; when 160 baked goods are sold, the expenses are paid. Any additional sales are profit.

You can also find the break-even point by setting P to 0:

(1)   \begin{equation*}0=1.50n - 240\end{equation*}

Add 240 to both sides:

(2)   \begin{equation*}240=1.50n\end{equation*}

Now, divide both sides by 1.50:

(3)   \begin{equation*}160=n\end{equation*}

If desired, we could add (160,0) to our table of values.

Finding the break-even point can be a common question in math 10 and math 11. Its attractiveness is its real-world meaning:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: solving exponential equations with logs

Tutoring math, your grade 12 students ask about logs.  The math tutor continues the discussion from last post, showing how to use logs to solve an exponential equation.

In my previous post, I showed how you might solve an exponential equation without logs when the sides can be brought to a common base.

Common base or not, you can always use logs to solve an exponential equation (provided it has a defined answer). Consider the following example:

Solve

(1)   \begin{equation*}13^{x+3}=7^{2.1x+5}\end{equation*}

To tackle this equation, the rule of logs that you need to know is

(2)   \begin{equation*}loga^b=bloga\end{equation*}

In other words, after you take the log, you can “bring down” the exponent to the front, where it becomes a multiple.

We take the log of each side of our equation:

(3)   \begin{equation*}log{13^{x+3}}=log{7^{2.1x+5}}\end{equation*}

We invoke the “exponent-to-multiple” rule for logs:

(4)   \begin{equation*}(x+3)log13=(2.1x+5)log7\end{equation*}

Apply the distributive law to each side:

(5)   \begin{equation*}xlog13+3log13=2.1xlog7+5log7\end{equation*}

Now, we use algebra to get all the x terms to one side, then all the constants to the other. Specifically, we subtract 2.1xlog7 from both sides, then subtract 3log13 from both sides, to get

(6)   \begin{equation*}xlog13-2.1xlog7=5log7-3log13\end{equation*}

Next, we factor out x from the left side:

(7)   \begin{equation*}x(log13-2.1log7)=5log7-3log13\end{equation*}

Finally, we divide both sides by (log13 – 2.1log7) to isolate x on the left. We arrive at the solution

(8)   \begin{equation*}x=\frac{(5log7-3log13)}{(log13-2.1log7)}\end{equation*}

If you carefully key the solution into a calculator, you’ll hopefully get -1.337334 (rounded). In order to ascertain that it truly is the solution, we substitute it for x in the original equation as follows:

(9)   \begin{equation*}13^{(-1.337334+3)}=7^{(2.1(-1.337334)+5)}\end{equation*}

You’ll get, rounded to four decimal places,

(10)   \begin{equation*}71.1403=71.1403\end{equation*}

Since the left side equals the right side, our solution is correct.

The logs method works whether or not the sides have a common base. Some people prefer it for every exponential equation they face.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: solving exponential equations

Tutoring math, you might be asked about these by your grade 12 precalculus students.  The math tutor opens the discussion….

There are two types of exponential equations:  those that don’t need logs to be solved, and those that do.  We’ll cover the ones that don’t need logs this time; next time, we’ll cover those that do.

Consider the following example:

Solve

(1)   \begin{equation*}25^{2x-3}=(\frac{1}{125})^{-x+7}\end{equation*}

Solution:

First of all, we notice that both sides are base 5 numbers: 25=52, while

(2)   \begin{equation*}\frac{1}{125}=5^{-3}\end{equation*}

(You may want to brush up on negative exponents; see my post here.) The fact that both sides are in the same base means we don’t need logs; otherwise, we would.

Since both sides are in the same base, we proceed as follows:

First, we rewrite both sides in the common base (in this case, 5):

(3)   \begin{equation*}(5^2)^{2x-3}=(5^{-3})^{-x+7}\end{equation*}

Recall the exponent-to-an-exponent law: (xa)b=xab. For example, (x2)5=x10. We apply that law to each side:

(4)   \begin{equation*}5^{4x-6}=5^{3x-21}\end{equation*}

Next, we invoke another exponent law: Ap=Aq→p=q. That is, if the bases are the same, and the left side equals the right side, the exponents must be equal. (Of course, there must be no other terms, nor any arithmetic operations pending, on either side.)

Therefore,

(5)   \begin{equation*}4x-6=3x-21\end{equation*}

We subtract 3x from both sides, then add 6 to both sides, yielding

(6)   \begin{equation*}x=-15\end{equation*}

If you have trouble believing -15 is the answer, here’s an option: Let’s take an impartial calculator and, substituting -15 for x, carefully enter the original left side:

(7)   \begin{equation*}25^{(2(-15)-3)}=7.37869763x10^{-47}\end{equation*}

Let’s do the same for the right side:

(8)   \begin{equation*}(\frac{1}{125})^{(-(-15)+7)}=7.37869763x10^{-47}\end{equation*}

Not necessarily everyday numbers; still, they’re equal:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

English: conjunctions: “ands, ifs, or buts”

Tutoring English or any subject, you recall your own school memories.  The English tutor brings forward the “ands, ifs, or buts” phrase, hoping to breathe in new life….

How many of you remember the teacher’s warning:  “I want this done by Monday, no ands, ifs or buts.”  If you’re in your forties and hail from the Maritimes, you heard it, whether directed at you or someone else.  It meant, or course, “no excuses.”

However rarely “ands, ifs, or buts” is used today, the phrase contains a ripe assortment of conjunctions.  Conjunctions are, or course, words used to join ideas to form sentences.

The most commonly used conjunction is and.  While it has its uses, it’s not exactly a mark-fetcher from an English teacher’s (or professor’s) point of view.  What can be used instead, that marks might stick to a little better?

I read once that but is logically equivalent to and.  Logically equivalent, perhaps – yet so much better in so many cases.  Consider the following:

1) Mother wants to make cupcakes and she needs more flour.

2) Mother wants to make cupcakes but she needs more flour.

The first sentence states the two facts as being virtually independent.  The second one establishes the need for more flour in order to make the cupcakes.  Therefore, the word but gives more meaning than and.

Consider another example:

3) I’ve narrowed the wall colour to a few possibilites, and I’m waiting for your opinion.

versus

4) I’ve narrowed the wall colour to a few possibilites, but I’m waiting for your opinion.

The two sentences may be logically equivalent, but 4) conveys the idea that the speaker won’t decide until they talk to you.

I read that “and” and “but” are logically equivalent in a computer science manual.  The idea immediately impressed me.  At first, I didn’t believe it.  However, as I explored the comparison between “and” and “but”, I realized that, from a certain point of view, the manual was correct.  Still, people don’t commonly sense “and” and “but” as being equivalent, which is why “but” can offer richer meaning than “and.”  When I’m about to use “and”, I usually ask myself if “but” will work better.  I use “and” much less today than I did ten years ago.

I’ll be saying much more about conjunctions in future posts.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: a word problem in two variables using substitution

Tutoring math, you notice practical problems that can be entertaining.  The math tutor presents one, highlighting the substitution method.

Example:  You have some oranges and a box.  When you weigh the box with three oranges in it, the mass is 1000g.  With ten oranges in it, the mass is 1770g.  Assuming the oranges are all the same, find the mass of an orange, as well as the mass of the box.

Solution:  We organize ourselves with a couple of “let” statements:

Let x=the mass of an orange.

Let y=the mass of the box.

Next, we translate The box with three oranges in it has mass 1000g into math, using our variables:

(1)   \begin{equation*}3x + y = 1000\end{equation*}

Similar for With ten oranges in it, the box has mass 1770g:

(2)   \begin{equation*}10x + y = 1770\end{equation*}

To solve the problem, an easy technique to use is substitution. To do so, we isolate y in the first equation (by subtracting 3x from both sides):

(3)   \begin{equation*}y = 1000 -3x\end{equation*}

Now, we substitute 1000 – 3x for y in the second equation:

(4)   \begin{equation*} 10x + 1000 - 3x = 1770\end{equation*}

We take our new equation and simplify it:

(5)   \begin{equation*}7x + 1000 = 1770\end{equation*}

Next, we subtract 1000 from both sides:

(6)   \begin{equation*}7x = 770\end{equation*}

Dividing both sides by 7, we get

(7)   \begin{equation*}x = 110\end{equation*}

Re-examining our let statements, we recall x is the mass of an orange. Now, to get y, we substitute 110 for x in either of our equations:

(8)   \begin{equation*}3(110) + y = 1000\end{equation*}

(9)   \begin{equation*}330 + y =1000\end{equation*}

Subtracting 330 from both sides, we get

(10)   \begin{equation*}y=670\end{equation*}

So, the mass of an orange is 110g, while the box is 670g.

The substitution method is slick when isolating one of the variables leaves you with whole numbers. Otherwise, you might prefer the elimination method, which I will cover in a future post:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Physics: calculating the normal force on a slope

Tutoring physics, you cover this every year.  The tutor offers a quick guide.

The normal force (FN) refers to the gravitational force perpendicular to the surface an object sits on. If the object sits on a level surface, the normal force is merely mass times gravity, or

(1)   \begin{equation*}F_N=mg\end{equation*}

where m is the mass in kg and g=-9.8N/kg on Earth’s surface.

Example: Calculate the normal force acting on a 1.2kg textbook lying flat on a table.

Solution: FN=mg=(1.2)(-9.8)=-11.76N or -12N in significant digits. The negative sign just means the force of gravity is downwards.

If the object is lying on a sloped surface, the normal force is no longer straight down; rather, it acts perpendicular to the angle of the slope. Trigonometry is required, in such a case, to calculate the normal force.

Example: Calculate the normal force acting on a 40.kg object lying on a 27º incline.

Solution: We are best off to draw a diagram:

Normal Force digram

Notice that, in the red triangle, FN is adjacent to the 27º angle. The hypotenuse is mg=-392. Therefore,

(2)   \begin{equation*}F_N=mgcos27^o=-392cos27^o=-349.3N\ =-350N\ in\ sig\ digs\end{equation*}

It only makes sense that, as the incline gets steeper, the normal force diminishes. The gravitational force shifts toward pulling the object down the incline instead.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Statistics: Standard deviation on the TI-30XA

Tutoring math or statistics, you need to be familiar with various common scientific calculators. The math tutor continues the standard deviation series with a look at how to do it on Texas Instruments’ TI-30XA.

Unlike the Sharp EL-520W or the Casio fx-260Solar, the TI-30XA does not require you to be in a certain mode to use the statistical functions. Similar to the Casio, however, the TI-30XA is reverse entry. Therefore, to enter -18, you key in 18, then + ↔ –.

Example: Find the standard devation of the data set {-18,16,0,45,100,32,27}.

Solution: First, you should clear any data already in there. To do so, press 2nd 7. (If the stats memory is already clear, you’ll get Error. Don’t worry; just press ONC.) Notice the yellow CSR above the 7; I believe it stands for “Clear Stats Registers”.

Enter each number, pressing the Σ+ button after each. After you enter -18, then press Σ+, you should see “n=1” across the screen. Then, key in 16 and press Σ+, and so on. If you make a mistake, you can pressONC, then enter the correct number and continue by pressing Σ+. If you want to remove an entry from the list later on, you can do so with 2nd Σ+. Noticing the yellow Σ- printed above the Σ+, you’ll get the idea.

Once you’ve entered the last number (in our case, 27), then pressed Σ+, it’s time to find the standard deviation. If you want σn, press 2nd ÷ (once again, see the yellow σxn above ÷). You should get 34.9, just like in the previous two articles. Otherwise, press 2nd √x to get σn-1. For Math Foundations 11, I believe σn is preferred.

Because the calculator I had in high school was similar to the TI-30XA, writing this article has been somewhat of a homecoming for me. I still appreciate the simple layout, with dedicated keys for square root and data entry.

Have a great weekend:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.