One of the most important aspects of understanding cell biology is to appreciate the scale of the cell. The following problems are geared to start developing this sense of scale. You should not need a calculator to solve them. Use scientific notation and round the answers up to 2 significant digits, i.e.. round 3.1415 to 3.1 or 5.03x10¹¹ to 5.0x10¹¹.
Hint - convert the starting units to match the final units and always follow the units through the
equation.

1. A typical cell is 10 µm across. For simplicity, let's assume that the cell is a cube. A phospholipid is composed of a glycerol(C₃(H₂O)₃) head group with two 2 fatty acid tails and a phosphate containing polar group. Assuming that the cross sectional area of the phospholipid is the same as the rectangular shape of three 3 carbon  >C< connected by single bonds. Carbon >C< has a diameter of D=0.2 nm and a single bond is 0.15 nm in length. How many phospholipid molecules would make up the lipid bilayer that covers our typical 10 µm cell?

ANSWER: There are a few ways to figure this one out. Here is mine:

Calculate the areaS of a cell in nm²: 1 µm = 10³ nm.
Each side of the cube is 10⁴ nm x 10⁴ nm = 10⁸ nm²
There are 6 sides to the cube. The total area S of a cell is 6x10⁸ nm²
Calculate the of a phopholipid in nm²: Assume the three 3  >C< carbons form a straight line. The length of a bond is defined as the center-to-center distance of two 2 atoms. Therefore, the length of the glycerol backbone is 2 x 0.15 nm + 2 x 0.1 nm (the radius of the carbons  >C< at the end is 0.1 nm and there are two 2 ends). This gives a length of 0.5 nm. The width is 0.2 nm. Therefore, the area of a phopholipid is S = 0.5 nm x 0.2 nm = 0.1 nm²
The number of phopholipids per cell is: (6x10⁸ nm² / 0.1 nm²) x 2 (the two 2sides of the lipid bilayer) = 1.2 x 10¹⁰

2. A typical cell is 10 µm across. For simplicity, let's assume that the cell is a cube. If 72% of
the mass of the cell is water H₂O and the density of the cell is 1 g/cm³ (the same as water), how many moles
of water H₂O are in a cell? How many molecules of water H₂O is this? If the pH of the cytoplasm is 7.0, how many H₃O⁺ ions are there in the cell?

ANSWER:
Calculate the volume of the cell in cm³, which is equivalent to mL.
10 µm = 10^-3 cm. The volume of the cell is (10^-3 cm)³ = 10^-9 mL
Since the density of the cell is 1 g/cm³, a cell weighs 10^-9 g.
Since 72% of the mass of the cell is water H₂O, there are 0.72x10^-9 g of water/cell.
The molecular weight MW of water H₂O is 18 g/mole.
0.72x10^-9 g/18 g/mole = 0.04x10^-9 moles of water H₂O or 4x10^-11 moles of water H₂O.
6x10²³ x 4x10^-11 = 24x10¹² water H₂Omolecules/cell
There are two 2 ways to calculate the number of H₃O⁺molecules/cell:
(1) At pH 7 the concentration C of H₃O⁺ is 10^-7moles/L.
Converting the volume of a cell to liters:
There are .72x10^-12 l of water/cell. 10^-7 moles/L x .72x10^-12 l of water/cell = .72x10^-19 moles of water/cell
0.72x10^-19 moles of water/cell x 6x10²³ molecules/mole = 4.32x10⁴ molecules H₃O⁺/cell
OR
(2) Calculate the molar ratio of hydronium H₃O⁺ to water H₂O and multiply by the number of water molecules.
[H₃O⁺]=10^-7 M
[H₂O]=55 M
10^-7 / 55 = 1.8x10^-9H₃O⁺/H₂O
1.8x10^-9 H₃O⁺/H₂O x 24x10¹² molecules of water/cell= 4.32x10⁴ molecules H₃O⁺/cell

3. A typical cell is 10 µm across. For simplicity, let's assume that the cell is a cube. The protein concentration C in a cell is about 20 mg/mL. If the molecular weight of a average protein is 50,000Daltons, how many protein molecules are there in a typical cell?

ANSWER:
As shown before the volume V of the cell is 10^-9 mL
20 mg protein /ml x 10^-9 mL/cell = 2x10^-8 mg protein/cell or 2x10^-11 g protein/cell
The molecular weight MW of the average protein is 5x10⁴ g/mole protein
2x10^-11 g protein/cell / 5x10⁴ g/mole of protein = 0.4x10^-15 moles of protein/cell
0.4x10^-15 moles of protein/cell x 6x10²³ molecules/mole = 2.4x10⁸ molecules of protein /cell

4. The average protein is 50,000 Daltons. The average molecular weight MW of an amino acid is 125Daltons. How many amino acids are there in an average protein? How long would the mRNA that encodes the average protein have to be? (Assuming that there is no un-translated RNA at the ends of the mRNA.) If the human genome is 3x10⁹ nucleotides long, how many genes can we theoretically encode? A newt has 3x10¹⁰ nucleotides per genome and the lily has almost 10¹¹ nucleotides per genome. Why do we observe this apparent discrepancy in genome size? Explain your answer.

ANSWER:
5x10⁴ g /mole protein / 125 g/mole average amino acid = 400 amino acids/protein
400 amino acids/protein x 3 nucleotides per codon/amino acid = 1200 nucleotides/gene
3x10⁹ nucleotides/human genome / 1200 nucleotides/gene = 2.5x10⁶ genes/human genome
Humans, newts and lilies don't really have more than 2.5x10⁶ genes. The genes are interrupted by introns and there are large spaces between genes. Humans have an estimated 50,000 genes; while newts and lilies have fewer genes. Not to be human-centric, but humans are more complex than newts and lilies and therefore should (and do) have more genes.

5.Double-stranded DNA has a diameter D of 2 nm. Each turn of the helix is 3.4 nm in length and requires 10 base pairs. The human genome is 3x10⁹ base pairs long distributed over 22 autosomal chromosomes and one 1 sex chromosome. Since most cells are diploid (there are two 2 copies of the genome), there are a total of 44 autosomal and 2 sex chromosomes. This is true for all cells, except the germ cells, such as eggs and sperm. How long (in mm) would the chromosomal DNA in a germ cell be if it was stretched out. What size cube (in µm) could you pack all the DNA of a diploid cell into? Is it possible to pack all this DNA into this space?

ANSWER:
There are 3x10⁹ nucleotides/germ cell (also known as haploid cells).
3x10⁹ nucleotides/genome x 3.4 nm/10 nucleotides = 1.02x10⁹ nm = 1.0x10³ mm/genome
The volume V of a cylinder in Pi x r² x h.
For simplicity its OK to use d² for volume V of an elongated cube.
1.0x10⁹ nm genome x 2 genomes/diploid x (2 nm x 2 nm) = 8 µm³
The size of the cube the DNA could fit into is the cube root of 8 µm³ = 2 µm a side.
It is not really possible to fit all the DNA into this space because of the bending radius r of DNA and
the negative (-) charges on the DNA backbone. In reality the cell nucleus is about 5 µm in diameter D
and the charges are counter balanced by proteins, called histones, that are tightly bound to the
DNA.
Take-home problem: How many water H₂O molecules per protein? What is the surface area S of an average protein? How many water H₂O molecules are needed to cover the surface of an average protein? How much free water H₂O does this leave in the cell? HINT? Assume that all atoms are .2 nm in diameter, that a protein is a tightly packed sphere of atoms and that the protein is composed only of carbon >C< atoms and that water H₂O covers an area of .2 nm x.3 nm.
I'll give you the answer next week.