[問題] 關於虎克定律的題目
題目來源:課本習題
瓶頸: 知道是力量作用於DNA上,但是不知道要怎麼解釋跟表達
It is possible to see with the aid of a powerful microscope that a long piece
of double-strand DNA is flexibility, with the distance between the ends of
the chain adopting a wide range of values.
This flexibility us important because it allows DNA to adopt very compact
conformations as it is packaged in a chromosome.
It is convenient to visualize a long piece of DNA as s freely jointed chain,
a chain of N small, rigid units of length ι that are free to make any angle
with respect to each other.
The length ι, the persistence length, is approximately 45 nm, corresponding
to approximately 130 base pairs.
You will now explore the work associated with extending a DNA molecule.
(a)Suppose that a DNA molecule resists being extended from an equilibrium,
more compact conformation with a restoring force Ｆ=－ｋfχ, where χ is the
difference in the end-to-end distance of the chain from an equilibrium value
and ΚF is the force constant.
Systems showing this behaviour are said to obey Hooke's law.
(i) What are the limitations of this model of the DNA molecule?
(ii)Using this model, write an expression for the work that mist be done to
extend a DNA molecule by χ. Draw a graph of your conclusion.
(b)A better model of a DNA molecule is the one-dimensional freely jointed
chain, in which a rigid unit of length ι can only make an angle of 0。 or 180
。 with an adjacent unit. In this case, the restoring force of a chain
extended by χ=nι is given by
ｋΤ １＋ｖ
Ｆ ＝ ______ ｌｎ［ ________ ］ ｖ＝ｎ／Ｎ
２ι １－ｖ
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where k=1.381×10 JK is Boltzmann's constant (not a force constant).
(i)What are the limitations of this model?
(ii)What is the magnitude of the force that must be applied to extend a DNA
molecule with Ν=200 by 90 nm?
(iii) Plot the restoring force against v, noting that v can be either
positive or negative.
How is the variation of the restoring force with end-to-end distance
different from that predicted by Hooke's law?
(iv)Keeping in mind that the difference in end-to-end distance from an
equilibrium value is ｘ＝ｎｌ and, consequently, ｄｘ＝ｌｄｎ＝Ｎｌｄｖ,
write an expression for the work of extending a DNA molecule.
(v)Calculate the work of extending a DNA molecule from v=0 to v=1.0.Hint.You
must integrate the expression for w.
The task can be accomplished easily with mathematical software.
(c)Show that for small extensions of the chain, when v<<1, the restoring
force is given by
ｖｋＴ ｎｋＴ
Ｆ≒ ————— ＝ —————
ι Ｎι
Hint. See Appendix 2 for a review of series expansions.
(d)Is the variation of the restoring force with extension of the chain given
in part (c) different from that predicted by Hooke's law?
Explain your answer.
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