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The mystery length of polymers: What is persistence length and how does it affect molecular behavior?
In polymer science, persistence length is a fundamental mechanical property used to quantify the bending stiffness of a polymer. This property makes polymer molecules behave like flexible elastic rods or beams. The behavior of a polymer differs significantly depending on its length: if a segment of the polymer is shorter than the persistence length, the segment will behave like a rigid rod; for segments longer than the persistence length, their properties can only be described in a statistical manner. Description, a model similar to a three-dimensional random walk.
The persistence length is defined as the distance over which the correlation of the molecular orientations disappears.
More formally, the persistence length P can be defined as the sum of the average projections of all subsequent bonds j (j ≥ i) of each bond i onto some segment of the chain in infinite length. Specifically, this can be found by considering a vector that is tangential to position 0 and then studies the angle θ at a distance L from position 0.
The expected value of the persistence length decays exponentially with distance. The formula is: ⟨cos θ⟩ = e^{-(L/P)}.
Typically, the persistence length P is taken to be half the Kuhn length, i.e. the length of the assumed freely connectable segment. The persistence length can also be expressed by the bending stiffness Bs, Young's modulus E and the cross-section of the polymer chain.
Taking into account the electrolyte shielding, the persistence length of the charged polymer will depend on the surrounding salt concentration. The Odijk, Skolnick and Fixman model was used to describe the persistence length of charged polymers.
Examples of persistence length
For example, the persistence length of a living spaghetti noodle has been estimated to be on the order of 1018 meters (assuming a Young's modulus of 5 GPa and a radius of 1 mm). The persistence length of double-helix DNA is about 390 angstroms. Just because a living spaghetti has such a large persistence length doesn't mean it's inflexible; it just means that at 300K, the spaghetti would need 1018 meters to overcome the Bends due to thermal fluctuations.
Take a long, slightly flexible rope, for example; over short distances the rope is essentially rigid. When you look at two points of the rope that are very close together, the directions of their motion are highly correlated. But if you choose two points on the rope that are far apart, then their tangent vectors may point in different directions. When we plot the tangential angle correlation between two points as a function of distance, we see a graph that is expected to be 1 (perfect correlation) at zero distance, and then decreases exponentially as distance increases. The persistence length is the characteristic length scale of this exponential decay.
Tools for measuring duration
The persistence length of single-stranded DNA can be measured using a variety of tools, most of which are based on the worm-like chain model. For example, single-stranded DNA is labeled at both ends with both applicator and acceptor dyes to measure the average end-to-end distance, which is reflected in the FRET efficiency. The FRET efficiency was then converted to persistence length by comparing it with the FRET efficiency calculated based on the worm-like chain model.
Some recent attempts have combined fluorescence correlation spectroscopy (FCS) with the HYDRO program. The HYDRO program is an upgrade of the Stokes-Einstein equations. This equation assumes that the molecules are purely spherical and calculates the diffusion coefficient, which is inversely proportional to the diffusion time. However, the HYDRO program is not limited by the shape of the molecule. The diffusion times of multiple worm-like polymers were generated, calculated using the HYDRO program, and compared with the experimental diffusion times of FCS to estimate the persistence length of single-stranded DNA and to find the optimal one by adjusting the polymer properties. value.
The persistence length of a polymer is not only a measure of its fundamental properties, it is also closely related to the characteristics and functions of biomolecules and their behavior in various environments. Have you ever wondered how these tiny molecules work in such wonderful ways in nature?
