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The Phys i c s o f Li f e Physics 171: The University of Oregon, Department of Physics Prof. Raghuveer Parthasarathy Department of Physics The University of Oregon Eugene, OR 97403-1274 Email: raghu@uoregon.edu

Examples  of  Worksheets   Contents  

Mechanical Similarity and Bone Shape pages 2-5

on Brownian Motion pages 6-7

DNA and some numbers page 8

Neurons and Diffusion page 9

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Mechanical  Similarity  and  Bone  Shape  

(i.e.  Why  can’t  elephants  jump?) Photo: Patrick Gries (http://www.theguardian.com/arts/gallery/2007/oct/26/photography?picture=331083586)

1. We’ve seen in class the femurs of an elephant and a dog. (Yes, this is awesome.) Note their lengths and diameters, and the factor by which the elephant’s measurement is greater than the dog’s.

Length (cm) Diameter (cm) Dog

Elephant

Factor

2. Plot on the logarithmic graph below Diameter vs. Length for each of these two animals

3. Suppose the dog’s bone grew isometrically. Draw a line on the graph indicating the diameter vs. length relationship that would result. What slope should your line have? Is the elephant data point on this line?

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Impala. (from Wikipedia.) Bovids.    We’ve seen in class the graph of bone diameter vs. length for the humerus (shoulder-elbow bone) of many species of African bovids, reprinted again at the end of this document. Let’s work through what it means...   4. Notice that bone width vs. bone length for a *lot* of different bovids is a straight line on a log-log plot. What is the scaling exponent for bone width as a function of bone length? 5. What exponent would we have for *isometric* scaling? [Draw a few cylinders with the same length and / different cross-sectional areas] 6. Here’s how the strength of a bone (i.e. the force it can support) scales with its size: 7. Suppose an organism *isometrically* doubles its width? By what factor would the gravitational force (pulling it downward) increase?

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8. Suppose an organism *isometrically* doubles its width? By what factor would the strength of its bones increase? 9. Based on #7 and #8, but without calculating exact numbers, what would you conclude needs to “happen” to the width of an animal's weight-bearing bones relative to their length if it is to grow larger? Answer in words, not numbers. (In addition to words, illustrate this with diagrams of cylinders.) 10. Now for numbers: How should bone diameter (non-isometrically) scale with length so that bone strength scales in the same way as the force of gravity? (I.e. so that both involve length raised to the same exponent?) 10. Why do we say that the bovids in the graph are “mechanically similar?” 11. Is the elephant mechanically similar to the dog? (Draw a line indicating “mechanical similarity” on your graph – is the elephant on it?) [I’ll show more animals’ measurements in class.] 12. Why can’t elephants jump?

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on  Brownian  Motion     First let’s think about Non-random (non-Brownian) motion: For example, consider an object moving with a constant velocity, such a car on a highway. 1 Fill in this chart, of how far cars with various speeds travel. Then plot position vs. time, on both a linear-axis and log-axis graphs.

Distance (Miles) Time (hrs.) Car @ 20 mph Car @ 50 mph Car @ 100 mph 1

2

40

100

200

3

4

2 Writing distance ∝ timep, what is p?

Now let’s think about Brownian Motion.

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We can think of Brownian motion as a “random walk,” as described in class. (Imagine, for example, deciding whether to take a step to the right or to the left based on a coin flip.) In class we’ll see graphs of simulated random walks. Let’s note a few things, denoting position by x and time by t. The notation <x> means the average value of x. (Imagine averaging over many walkers.) 3 Sketch what x vs. t looks like for a few random walkers (as shown in class): 4 What is <x> for a randomly moving, “Brownian” walker? Does it change over time? 5 Suggest a property of our walks that we could measure that, on average is not zero. The class will suggest things; sketch them here. Do any show particular scaling behavior? 6 What relationship between distance and time can we infer about Brownian Motion? Also: write it in a few different ways (that we’ll show in class). Note (from class) what the term “diffusion coefficient” means. 7 If a Brownian particle travels 10 µm in 1 second, how long would it take to travel 20 µm?

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DNA  and  some  numbers   DNA-related numbers to remember1

• Number of base pairs in the human genome (i.e. the number of “rungs” in DNA in each of your cells): 3 × 109. (Can just remember 109)

• Number of genes in the human genome: About 30,000 . • Total length of DNA in each of your cells: 1 meter • Width of a DNA molecule: about 1 nm, i.e. 1×10-9 m. (It’s actually 2 nm,

but we needn’t be exact. Other numbers to remember:

• The typical size of a bacterium: ≈ 1 µm. • The size of a typical animal cell: ≈ 10 µm.

Other numbers you don’t have to remember:

• The number of base pairs needed to encode one amino acid of a protein: 3. • The diameter of the nucleus of a typical animal cell is about 1 µm. • Proteins are chains of amino acids. Average length of a protein: about 500 amino acids. • The persistence length of DNA is about 50 nm.

  Coding  and  Non-­‐Coding  DNA.       (i) How many base pairs are needed to encode a typical protein? (ii) How many base pairs are needed to encode all the proteins in the human genome? (iii) Is this roughly equal to 3 × 109 ? Speculate on what this means.

1 As you know, I’m strongly opposed to memorizing things. However, DNA is so important, and DNA-related issues are so common in modern society, that it’s worth keeping a few facts in mind. As you’ll see, these few facts reveal a lot.

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Neurons  and  Diffusion    What se ts the speed o f your thoughts? This is a complex question, but a key part of it involves the time neurotransmitters need to diffuse across the junction between neurons in “chemical synapses.” Let’s figure this out. A glass bead with a 1 µm diameter has a “diffusion coefficient” (the slope of <distance2> vs. time) of 0.1 µm2/s. 1 Draw a graph of 2distance vs. time for this bead; label your

axes.

National Institutes of Health / Wikimedia

/http://en.wikipedia.org/wiki/ File:Chemical_synapse_schema_cropped.jpg

A typical neurotransmitter, like acetylcholine, has a diameter of 1 nm (10-9 m). 2 What’s its diffusion coefficient? 3 Draw a graph of 2distance vs. t for this neurotransmitter; label your axes.

4 The size of a synaptic cleft is about 10 nm. How long does it take the neurotransmitter to travel this distance?