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We study the behaviors of mean end-to-end distance and specific heat of a two-dimensional intrinsically curved semiflexible biopolymer with a hard-core excluded volume interaction. We find the mean square end-to-end distance
Temperature (T) seriously affects the property of a polymer. It is well known that a flexible polymer has three different phases below, at, or above the θ-temperature.[1] The θ-temperature is the temperature at which the phase transition takes place and it provides a clear border to separate these phases. At the θ-temperature, the excluded volume effect (EVE) can be ignored or the EVE is balanced by the attractive interactions between polymer segments and solvent molecules, the conformation of a polymer can be modeled as a random walk of monomer subunits, and such a polymer is referred to as an ideal chain, a free-jointed chain, or a Gaussian chain. Correspondingly, solvents at the θ-temperature are called θ-solvents.[1] Moreover, in good solvent or above the θ-temperature, EVE dominates the conformation of a polymer so that the chain is in a swollen state and can be described by a self-avoiding random walk (SAW).[1–7] In contrast, in poor solvent or below the θ-temperature, the attractive forces between different segments suppress the excluded volume effect so that the polymer chain collapses into a compact globule state. Temperature also greatly affects the mechanical property of a polymer. For swollen or ideal polymer chains, the polymer extends progressively with increasing force so that no sharp transition in extension occurs.[8–11] However, in poor solvent, the extension of a polymer is subjected to a first-order transition.[12–14]
Biopolymers can be classified as flexible and semiflexible. A flexible biopolymer means that to bend it does not cost any energy. It follows that its ground-state configuration (GSC, or the configuration with the lowest energy) is not unique so that there is no way to define an intrinsic curvature (IC), since a finite IC means that the GSC must have a well-defined curvature. However, many biopolymers, such as DNA, actin, and microfilament, are rigid or semiflexible. To bend these biopolymers locally subjects them to an energy penalty. The conformal and mechanical properties of semiflexible biopolymers have attracted lots of attention owing to their importance in understanding the structure and function of bio-materials. The simplest model for a semiflexible biopolymer is the wormlike chain (WLC) model and it describes well the entropic elasticity of an intrinsically straight semiflexible biopolymer.[15–18] The GSC of the WLC model is unique and is a straight line, and at finite T, a WLC extends gradually with increasing stretching force or there is no sharp transition in extension. However, intrinsically curved biopolymers are also ubiquitous. For instance, special sequence orders favor a finite IC for some short dsDNA chains.[19–22] It has also been reported that a long-range correlated dsDNA has a macroscopic (intrinsic) curvature so that the WLC model fails to account for its property.[23]
Recently, we found that a finite IC alone can induce a discontinuous transition in extension[24,25] for a two-dimensional (2D) long semiflexible biopolymer. The IC plays a similar role as an attractive force in a poor solvent for a polymer and the bending rigidity plays the role of the strength of the attractive force. With a hard-core excluded volume interaction (EVI) and free of an external force, this system is the same as the SAW model if its bending rigidity is zero or at a very high temperature so the entropy dominates the conformation. On another limit, i.e., at a very large bending rigidity or at very low temperature, its conformation should be similar to an Archimedean spiral. The end-to-end distance of an Archimedean spiral is proportional to the square-root of its length. It is then intriguing to ask for such a system is there a temperature similar to the θ-temperature? In this work, we provide an answer to this question.
Moreover, recent progresses in experimental techniques make it possible to perform experiments on semiflexible biopolymers in a two-dimensional (2D) environment so that the property of biopolymers in 2D has attracted growing interest.[23–36] Therefore, for simplicity, we focus on the 2D system in this work.
The paper is organized as follows. In Section
A semiflexible biopolymer is often modeled as a filament. In a 2D case, the configuration of a filament is determined by a vector
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The continuous model does not include the excluded volume interaction (EVI). However, it is unreasonable to ignore the EVI in a 2D system. To find an exact result is very difficult when we consider the EVI, so that we discretize the continuous model and perform Metropolis Monte Carlo simulation[6] to study it in the off-lattice system. In the discrete model, a filament consists of N straight segments of length l0 joined end to end. The coordinates of the two ends of the i-th segment are therefore {xi−1,yi−1} and {xi,yi}. By replacing
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The thermal average, 〈B〉, of a physical quantity B(ϕ1,ϕ2,…,ϕN−1) is defined as the average with Boltzmann weight over all possible conformations, i.e.,
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In Metropolis Monte Carlo simulation, we equilibrate every sample from 106 to 2 × 106 Monte Carlo steps (MCS) before performing the averaging. The thermal average for a sample is taken from 2 × 107 to 3 × 107 MCS. Moreover, we take N = 50, 100, 200, 300, and 450 to examine the finite size effect. The range of κ is from 0 to 60. Note that the lp = 2κ of a double-stranded DNA is about 50 nm. c = 0.1, 0.2, and 0.5. We do not consider a larger c because it is impractical. We use the hinged boundary condition (BC) in simulation, i.e., we do not fix ϕ1 and ϕN.
The thermodynamical limit in this work means that we keep both l0 and c as constants but let N → ∞ so that the total length L = Nl0 → ∞. At a finite temperature, the fluctuation can suppress sharp transition, especially for a finite-size system. Therefore, to examine the occurrence of phase transition, one often has to evaluate the finite size effects. For this purpose, we evaluate the mean square end-to-end distance
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Exactly, in simulation we calculate 𝒞N = SN/kB. We also monitor
The mechanical property of the model in absence of EVI has been studied.[24,25,35] It finds that a nonvanishing c can induce a discontinuous change in extension for a semiflexible biopolymer. At T = 0, the transition is multiple-step and accompanied by unwinding loops, regardless of κ and L. However, a finite temperature represses the transition so that the discontinuous transition becomes one-step, requires sufficient large c and κ, and probably occurs only in the thermodynamical limit. However, a full picture of the thermal property of the model is not yet available.
When Uevi = 0, the
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Clearly
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It has a maximum value at
Because EL is in a quadratic form when κ > 0, from the equipartition theorem, it is straightforward to know that εL = kBT/2 and it follows 𝒞L = 1/2. In other words, both ε and 𝒞L are independent of κ and
At first, since we use a simple hard-core EVI, we expect that at a large κ or low T, the conformation of the system should tend to be close to an Archimedean spiral. In other words, the end-to-end distance of an Archimedean spiral gives a lower bound of
Meanwhile, because we adopt the hinged BC, the system has a rotational invariance so that exactly 〈xN〉 = 0 and
Moreover, from Figs.
![]() | Fig. 3. (color online) ![]() |
![]() | Fig. 4. (color online) A vs. κ when c = 0.2 and Uevi ≠ 0. The symbols are the same as those in Fig. |
![]() | Fig. 5. (color online) A vs. k when c = 0.5 and Uevi ≠ 0. The symbols are the same as those in Fig. |
From Figs.
However, also from Figs.
![]() | Fig. 6. (color online) A′ vs. κ when c = 0.1 and Uevi ≠ 0. The black solid line is given by A′ = 1. The other symbols are the same as those in Fig. |
![]() | Fig. 7. (color online) A′ vs. κ when c = 0.2 and Uevi ≠ 0. The black solid line is given by A′ = 1. The other symbols are the same as those in Fig. |
Furthermore, we calculate log
From Figs.
On the other hand, we do not find a compact globule state, i.e., a state with β = 0.5 like that found in the poor solvent for a flexible polymer.[7] This should be because we adopt a simple hard-core EVI which gives a lower bound of β, i.e., β ≥ 1. However, we also find that it is possible to have A′ < 1, i.e., to have a more compact conformation than a random walk. Therefore, a compact globule state is still possible if we adopt a relative soft-core EVI and it deserves a further investigation.
Finally, the elastic modulus under a uniaxial force can be found from[24,35]
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When κ = 0, in this system we have exactly εN = 0 so that 𝒞N = 0, and our simulation confirms this result, as we can see from Figs.
![]() | Fig. 10. (color online) 𝒞N vs. κ (1/κ in inset) when c = 0.1. The solid black line is given by 𝒞N = 0.2κ. The other symbols are the same as those in Fig. |
![]() | Fig. 11. (color online) 𝒞N vs. κ (1/κ in inset) when c = 0.2. The solid black line is given by 𝒞N = 0.2κ. The other symbols are the same as those in Fig. |
![]() | Fig. 12. (color online) 𝒞N vs. κ (1/κ in inset) when c = 0.5. The solid black line is given by 𝒞N = 0.2κ. The other symbols are the same as those in Fig. |
Finally, we do not find any clear peak in 𝒞N in all cases, or 𝒞N is a smooth function of κ or T. It therefore supports the conclusion obtained from
In summary, we study the thermal properties of a two-dimensional intrinsically curved semiflexible biopolymer with a simple hard-core EVI. We find that it always has
We focus on a constant intrinsic curvature in the present work, but we expect that the main conclusion is still valid for a 2D semiflexible biopolymer with short-range correlation in intrinsic curvatures. Moreover, we also expect that it is possible to observe similar phenomena in some constrained systems, such as in a flat box. We adopt a simple hard-core EVI in this work and it may prevent β < 1 so that a filament with a soft-core EVI deserves a further investigation.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] |