Cells, Membranes & Exchange — Problem set, Grade 10

All living things are made of one or more cells; the cell is the basic unit of structure and function in life; and all cells arise from pre-existing cells.

Cell theory, built up from Hooke, Leeuwenhoek, Schleiden, Schwann, and Virchow, makes three claims: (1) all organisms are composed of one or more cells, (2) the cell is the basic unit of structure and function, and (3) every cell comes from a cell that already existed (Virchow's 'omnis cellula e cellula').

400×.

Total magnification = eyepiece × objective = 10 × 40 = 400×. This means the image appears 400 times larger than the object's actual size. (Remember: magnification is separate from resolution, which sets the real limit on visible detail.)

Prokaryotic cells have no membrane-bound nucleus or organelles; eukaryotic cells do.

A prokaryote (bacterium or archaeon) keeps its DNA loose in the cytoplasm and has no membrane-bound organelles. A eukaryote seals its DNA in a true nucleus and compartmentalizes its work into membrane-bound organelles such as mitochondria, ER, and Golgi. (Eukaryotic cells are also typically much larger.)

Nucleus — stores DNA and directs the cell; mitochondrion — releases energy (ATP) through respiration; ribosome — builds proteins; Golgi apparatus — modifies, packages, and ships proteins.

The nucleus is the control center holding the DNA. Ribosomes read the instructions and assemble proteins. Those proteins are finished in the ER and then sent to the Golgi apparatus, which tags, packages, and ships them in vesicles. The mitochondrion powers all of this by releasing energy from food as ATP.

Ribosome (on rough ER) → rough ER → Golgi apparatus → vesicle → cell membrane → outside.

A ribosome on the rough ER builds the protein and threads it into the ER to be folded. The protein travels to the Golgi apparatus, which modifies and packages it into a vesicle. That vesicle moves to the cell membrane and fuses with it (exocytosis), releasing the protein outside the cell.

Because each phospholipid has a hydrophilic head and hydrophobic tails, and a bilayer lets the heads face water while the tails hide from it.

A phospholipid is amphipathic: its phosphate head is hydrophilic (water-loving) and its fatty-acid tails are hydrophobic (water-fearing). Since a cell is watery inside and out, the molecules form two layers with heads pointing outward toward the water on both sides and tails tucked together in the middle. This is the lowest-energy arrangement, so it happens spontaneously.

It describes the membrane as a fluid phospholipid sheet with proteins embedded throughout like tiles; those proteins, plus the oily core, make it selectively permeable.

The membrane is 'fluid' because its phospholipids drift sideways, and a 'mosaic' because many proteins are embedded in it. Small nonpolar molecules dissolve straight through the hydrophobic core, while large or charged molecules can cross only through specific transport proteins. That combination is what lets the membrane control precisely what enters and leaves — selective permeability.

(a) In pure water it swells and may burst; (b) in salt water it shrivels.

Osmosis moves water toward the side with more solute. (a) Pure water is hypotonic (less solute outside), so water rushes into the cell; with no wall to resist, the cell swells and can burst (lysis). (b) Concentrated salt water is hypertonic (more solute outside), so water leaves the cell and it shrivels (crenates).

Active transport.

The ions are moving from low to high concentration — against their concentration gradient. Passive transport can only move substances down a gradient (toward equal concentrations), so moving them the other way requires energy (ATP) and protein pumps. That is the definition of active transport, as seen in the sodium-potassium pump.

The 2-unit cell has a ratio of 3; the 4-unit cell has a ratio of 1.5. The smaller cell exchanges materials more efficiently.

For a cube, surface area = 6s² and volume = s³, so SA:V = 6/s. For s = 2: SA = 6×4 = 24, V = 8, ratio = 24/8 = 3. For s = 4: SA = 6×16 = 96, V = 64, ratio = 96/64 = 1.5. The smaller cell has more membrane surface per unit of volume, so it can move materials in and out fast enough to serve its interior — which is a major reason cells stay small or divide instead of growing huge.

15 µm

1.5 mm / 100 = 0.015 mm = 15 µm.

Absence from one view is weak evidence. The nucleus may be out of the focal plane, unstained and therefore low in contrast, or outside the section if the slide is a thin slice. A supported conclusion requires focusing through the specimen, using an appropriate stain, and comparing several cells; a known nucleated control on the same slide would strengthen it further.

Response criteria

• State a biologically accurate conclusion.
• Show the data, mechanism, or calculation that supports it.
• Include units, conditions, or limits where relevant.

The estimate is 2.4×25 = 60 μm. A magnification label describes a particular display enlargement; resizing changes it, while an image and scale bar resized together preserve the specimen-length ratio.

The estimate is 2.4×25 = 60 μm. A magnification label describes a particular display enlargement; resizing changes it, while an image and scale bar resized together preserve the specimen-length ratio.

Response criteria

• Use the stated measurements or conditions, with correct units.
• Connect the result to the biological mechanism or evidence relationship.
• Identify the assumption, model limit or further evidence requested.

The cell most likely secretes protein in quantity, such as a gland or pancreatic cell. Extensive rough endoplasmic reticulum indicates heavy protein synthesis and folding, a prominent Golgi indicates substantial modification and sorting, and vesicles at the surface indicate active export. The inference rests on the proportions of organelles present, not on any single structure, and would be confirmed by identifying the secreted protein.

Response criteria

• State a biologically accurate conclusion.
• Show the data, mechanism, or calculation that supports it.
• Include units, conditions, or limits where relevant.

8

1,200 / 150 = 8 times as many, consistent with a much higher sustained ATP demand.

Gene information is transcribed and used by ribosomes; selected secretory proteins enter the ER, undergo processing/sorting through Golgi and travel in vesicles to membrane fusion. A cytosolic enzyme generally follows a different localization pathway rather than compulsory secretion.

Gene information is transcribed and used by ribosomes; selected secretory proteins enter the ER, undergo processing/sorting through Golgi and travel in vesicles to membrane fusion. A cytosolic enzyme generally follows a different localization pathway rather than compulsory secretion.

Response criteria

• Use the stated measurements or conditions, with correct units.
• Connect the result to the biological mechanism or evidence relationship.
• Identify the assumption, model limit or further evidence requested.

Being small and nonpolar, it dissolves in the bilayer's interior and diffuses across membranes without a transport protein. The consequence is that it is not restricted to tissues with a specific transporter: it can enter most cells, including ones it was not intended for, and can cross barriers such as the one protecting the brain. Specificity must then come from what it binds inside, not from how it enters.

Response criteria

• State a biologically accurate conclusion.
• Show the data, mechanism, or calculation that supports it.
• Include units, conditions, or limits where relevant.

1

A 9 nm length exceeds the 8 nm thickness, so it can span the bilayer and project slightly from both faces, as transmembrane proteins do.

Charge and lipid-interior interactions limit the direct route; the selective channel changes the available pathway. Fluidity concerns component motion, not unrestricted permeability to every substance.

Charge and lipid-interior interactions limit the direct route; the selective channel changes the available pathway. Fluidity concerns component motion, not unrestricted permeability to every substance.

Response criteria

• Use the stated measurements or conditions, with correct units.
• Connect the result to the biological mechanism or evidence relationship.
• Identify the assumption, model limit or further evidence requested.

0 %

The solution is isotonic, so water enters and leaves at equal rates and the net rate is zero. Individual molecules still cross in both directions.

Mass gain means water entered, so that solution is hypotonic to the potato cells; mass loss means water left, so that solution is hypertonic. The solution producing almost no change is close to isotonic with the cells, which gives an estimate of the cells' internal solute concentration. The estimate is only as precise as the spacing of the concentrations tested, so intermediate concentrations and replicate strips would narrow it.

Response criteria

• State a biologically accurate conclusion.
• Show the data, mechanism, or calculation that supports it.
• Include units, conditions, or limits where relevant.

The change is −3/12×100 = −25%. Net water loss is consistent with a hypertonic external solution under the model. Comparable blotting, membrane integrity and limits on solute loss are needed before assigning all mass change to water.

The change is −3/12×100 = −25%. Net water loss is consistent with a hypertonic external solution under the model. Comparable blotting, membrane integrity and limits on solute loss are needed before assigning all mass change to water.

Response criteria

• Use the stated measurements or conditions, with correct units.
• Connect the result to the biological mechanism or evidence relationship.
• Identify the assumption, model limit or further evidence requested.

Many cells share the work, while specialized exchange and transport structures move materials between the environment and cells.

2

Surface area = 6 × 3² = 54; volume = 3³ = 27; ratio = 54/27 = 2.0 per unit length.

Volume grows with the cube of length while surface area grows with the square, so a larger cell has proportionally less membrane per unit of cytoplasm to supply. Beyond some size, diffusion across the surface cannot meet the interior's demand, so cells stay small. Folded or projecting surfaces work around the limit: microvilli on an intestinal cell, or the inner folds of a mitochondrion, add membrane area without adding volume.

Response criteria

• State a biologically accurate conclusion.
• Show the data, mechanism, or calculation that supports it.
• Include units, conditions, or limits where relevant.

At 2 mm: area 24 mm², volume 8 mm³, ratio 3 mm⁻¹. At 4 mm: area 96 mm², volume 64 mm³, ratio 1.5 mm⁻¹. Total surface increases while area per volume falls; actual exchange also depends on pathways and demand.

At 2 mm: area 24 mm², volume 8 mm³, ratio 3 mm⁻¹. At 4 mm: area 96 mm², volume 64 mm³, ratio 1.5 mm⁻¹. Total surface increases while area per volume falls; actual exchange also depends on pathways and demand.

Response criteria

• Use the stated measurements or conditions, with correct units.
• Connect the result to the biological mechanism or evidence relationship.
• Identify the assumption, model limit or further evidence requested.