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2. Spring constant of DNA. An "optical tweezer" apparatus (see 80 Appendix) can be used to measure the force needed to pull on a molecule to extend its length by a measurable amount. A measurement obtained for double-stranded DNA (dsDNA) is shown 60 in the graph, where the horizontal axis is labeled in micrometers, Overstretching 10 6 m, and the vertical axis in piconewtons, 10712 N. "Extension" transition Force (PN) refers to the distance the DNA has stretched from its relaxed state. 40 The "B-form" dsDNA is the type that most often exists under normal physiological conditions in the cell. We are going to model what 20 B form contour happens to DNA as it is stretched, trying to account for the different length behaviors of the DNA we observe in the graph. We'll consider the different regions of the graph separately. 10 15 20 25 30 For each of the following regions, show how you calculate the spring DNA extension (um) constant of the DNA in that region using the information from the graph. a) The region where the applied force is less than a few pN b) The region where the applied force is between a few and 65 pN (the "B-form" region) c) The region where the applied force is between 65 and 70 pN (the "overstretched" region) d) The region where the applied force is greater than 70 pNThere are two questions: 1. Muscles as springs, and 2. Spring constant of DNA. 1. Muscles as springs. The muscles in animals are complex structures consisting of bundles of long fibers that contract in response to chemical signals. The force provided by these contractile fibers when they are lengthened and when they are bundled depends on the physics of how connected springs add their forces. in order to understand how this works, let's consider a simplified model of a muscle: many springs connected in different ways. Let's think about how springs respond to being pulled rather than to how they pull on other objects. We'll take as our basic element a single spring (model of a fiber) with rest length {'0 and spring constant k. If it is pulled from opposite directions by a tension force, T, it will stretch an amount A! that satisfies Hooke's law T = km}, where the total stretched length of the spring is now [0 + At'. First let's consider the effect of linking fibers together end to end into a longer fiber. (In a biological muscle, multiple cells combine, creating single long cells with multiple nuclei.) This kind of connection is referred to as "series\". Let's start by considering two identical springs each having spring constant k, linked together (shown by the black oval in the figure) and NW 7' pulled from opposite ends by equal tension forces T. We imagine that they are connected by molecules that are short and don't stretch significantly compared to the springs themselves. 3) Consider this combination as a single "effective" spring. If one spring stretches a distance 113 when pulled by the force T, what distance AL would this combined spring stretch when pulled by the same force T? Explain your answer. Al." lAt b) If we define the effective spring constant of the combined springs by the equation T = keffAL, where AL is the total amount the series springs stretch (your answer to part a), how is kg\" related to k? Explain your answer. T: MiAi T\" KAt c) Now suppose that we have attached not just two springs in series, but N springs. Write an equation that expresses the effective spring constant of the combination using the spring constant of the original spring k and the number of springs N. d) Explain whether the combination of many springs connected in series is softer, stronger, or the same as a single spring. (Here, softer means easier to stretch and stronger means harder to stretch.) Next, let's consider the effect of linking fibers to the same connecting point on either end. This kind of connection is referred to as "parallel". Consider the same two identical springs each having spring T constant k, but this time with both linked at each end to the same object. Equal tension forces T are applied to the objects connecting the springs together. e) Again we will consider this combination as a single effective spring. Draw a free-body diagram for the plate at one end to which the two strings are attached. f) If one spring stretches a distance Af when pulled by the force T, what distance AL would this combined spring stretch when pulled by the same force T? Explain your answer.g) Now suppose that we have attached not just two springs in parallel, but NV springs. Write an equation that expresses the effective spring constant of the combination using the spring constant of the original spring k and the number of springs N. h) Explain whether the combination of many springs connected in parallel is softer, stronger, or the same as a single spring. () In the human body, the distance that muscles can stretch is limited by the size and range of motion of the body. If we assume that the maximum stretching distance of a muscle is fixed, then which is more desirable for the ability to lift heavier objects: a large effective spring constant or a small effective spring constant