# Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Derivation of Energy Functions

#### 2.2. Mathematical Preliminaries

#### 2.2.1. Combinatorial Objects

#### 2.2.2. A Bijection Between ${\mathfrak{T}}_{n}$ and ${\mathfrak{M}}_{n-1}^{2}$

- If e is the leftmost edge off a non-root node of down degree at least 2, assign the label U.
- If e is the rightmost edge off a non-root node of down degree at least 2, assign the label D.
- If e is the only edge off a non-root node of degree 1, assign the label I.
- If e is an edge off the root node, or if e is neither the leftmost nor the rightmost edge off its parent node, assign the label H.

#### 2.2.3. Markov Chains

- Irreducibility: For any $x,y\in \mathsf{\Omega}$, there is some integer $t\in \mathbb{N}$ for which ${P}^{t}(x,y)>0$.
- Aperiodicity: For any state $x\in \mathsf{\Omega}$, we have $gcd\{t\in \mathbb{N}:{P}^{t}(x,x)>0\}=1$.

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

#### 2.2.4. Coupling

- Each chain ${\left({X}_{t}\right)}_{t=0}^{\infty}$ and ${\left({Y}_{t}\right)}_{t=0}^{\infty}$, when viewed in isolation, is a copy of $\mathcal{M}$, given initial states ${X}_{0}=x$ and ${Y}_{0}=y$.
- Whenever ${X}_{t}={Y}_{t}$, we have ${X}_{t+1}={Y}_{t+1}$.

**Lemma**

**4.**

**Theorem**

**1.**

#### 2.2.5. Decomposition

**Theorem**

**2.**

**Theorem**

**3.**

**Lemma**

**5.**

- 1.
- $P(x,y)\ge a$ for all $x,y$ such that $P(x,y)>0$.
- 2.
- $\pi \left({\partial}_{i}\left({\mathsf{\Omega}}_{j}\right)\right)\ge b\pi \left({\mathsf{\Omega}}_{j}\right)$ for all $i,j$ with $\overline{P}(i,j)>0$.

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

- If heads, set ${Y}_{t+1}={Y}_{t}$. Let l be either 1 or $-1$, each with probability $1/2$. If possible, let ${X}_{t+1}={X}_{t}+l$ with probability $\frac{1}{2}min\left(\right)open="\{"\; close="\}">1,\frac{\pi ({X}_{t}+l)}{\pi \left({X}_{t}\right)}$. Otherwise, let ${X}_{t+1}={X}_{t}$.
- If tails, set ${X}_{t+1}={X}_{t}$, and update ${Y}_{t+1}$ the same way as we did for ${X}_{t+1}$ in the previous case.

## 3. Results

#### 3.1. Our Markov Chain on ${\mathfrak{M}}_{m}^{2}$

- If $l=1$, pick a random pair of consecutive symbols in ${X}_{t}$, and call this pair s. If s is $UD$ or $HH$, let ${s}^{\prime}$ be either $UD$ or $HH$ with probabilities $\frac{1}{1+{e}^{-\alpha}}$ and $\frac{{e}^{-\alpha}}{1+{e}^{-\alpha}}$ respectively. Let y be the string ${X}_{t}$ with s replaced by ${s}^{\prime}$. Otherwise, let $y={X}_{t}$.
- If $l=2$, pick i uniformly from $\{1,\cdots ,m\}$. If ${X}_{t}\left(i\right)$ is H or I, choose a symbol c to be either H or I with probabilities $\frac{{e}^{-\alpha}}{{e}^{-\alpha}+{e}^{-\beta}}$ and $\frac{{e}^{-\beta}}{{e}^{-\alpha}+{e}^{-\beta}}$ respectively. Let y be the 2-Motzkin path given by changing the symbol in ${X}_{t}\left(j\right)$ to c. Otherwise, we let $y={X}_{t}$.
- If $l=3$, pick i and j each uniformly from $\{1,\cdots ,m\}$. If each of ${X}_{t}\left(i\right)$ and ${X}_{t}\left(j\right)$ are either U or D, let y be the string ${X}_{t}$ with the symbols at indices i and j swapped. Otherwise, let $y={X}_{t}$.
- If $l=4$, pick a random pair of consecutive symbols in ${X}_{t}$, and call this pair s. If s is of the form $ab$ or $ba$ for some $a\in \{U,D\}$ and $b\in \{H,I\}$, let ${s}^{\prime}$ be the reverse of s, and let y be the string ${X}_{t}$ with s replaced by ${s}^{\prime}$. Otherwise, let $y={X}_{t}$.

Algorithm 1: The main Markov chain algorithm. This pseudocode calculates ${X}_{t}$ given ${X}_{0}$. |

Require: ${X}_{0}$ is a valid 2-Motzkin path of length m. |

$x\leftarrow {X}_{0}$ |

for $s=1\to t$ do |

$y\leftarrow x$ |

$l\leftarrow \mathrm{randInt}(1,4)$ |

if $l=1$ then |

$i\leftarrow \mathrm{randInt}(1,m-1)$ |

if $x[i:i+1]=UD$ and $\mathrm{Ber}\left(\right)open="("\; close=")">\frac{{e}^{-\alpha}}{2(1+{e}^{-\alpha})}$then |

$y[i:i+1]\leftarrow HH$ |

else if $x[i:i+1]=HH$ and $\mathrm{Ber}\left(\right)open="("\; close=")">\frac{1}{2(1+{e}^{-\alpha})}$then |

$y[i:i+1]\leftarrow UD$ |

else if $l=2$ then |

$i\leftarrow \mathrm{randInt}(1,m)$ |

if $x\left[i\right]=I$ and $\mathrm{Ber}\left(\right)open="("\; close=")">\frac{{e}^{-\alpha}}{2({e}^{-\alpha}+{e}^{-\beta})}$then |

$y\left[i\right]\leftarrow H$ |

else if $x\left(i\right)=H$ and $\mathrm{Ber}\left(\right)open="("\; close=")">\frac{{e}^{-\beta}}{2({e}^{-\alpha}+{e}^{-\beta})}$then |

$y\left[i\right]\leftarrow I$ |

else if $l=3$ then |

$i\leftarrow \mathrm{randInt}(1,m)$ |

$j\leftarrow \mathrm{randInt}(1,m)$ |

if ($x\left[i\right]\in \{U,D\}$ and $x\left[j\right]\in \{U,D\}$) and $\mathrm{Ber}\left(\right)open="("\; close=")">\frac{1}{2}$ then |

$y\left[i\right]\leftarrow x\left[j\right]$ |

$y\left[j\right]\leftarrow x\left[i\right]$ |

if y is not a valid 2-Motzkin path then |

$y\leftarrow x$ |

else if $l=4$ then |

$i\leftarrow \mathrm{randInt}(1,m-1)$ |

if ($x\left[i\right]\in \{U,D\}$ and $x[j+1]\in \{H,I\}$) or ($x\left[i\right]\in \{H,I\}$ and $x[j+1]\in \{U,D\}$) and $\mathrm{Ber}\left(\right)open="("\; close=")">\frac{1}{2}$ |

then |

$y[i:i+1]\leftarrow x[j+1]+x\left[j\right]$ |

$x\leftarrow y$ |

return x |

Algorithm 2: Algorithm to convert a sampled 2-Motzkin path to a plan tree. The pseudocode calculates ${\mathrm{\Phi}}^{-1}\left(x\right)$. |

Require: x is a valid 2-Motzkin path of length m. |

root ← new Node() |

// u will be where a new node will be added for an H or D symbol |

$u\leftarrow $ root |

// v will be always the last node added |

$v\leftarrow $ new Node() |

// the stack will keep track of previous values of u |

stack = new Stack() |

root.children.append(v) |

for $i=1\to m$ do |

node ← new Node() |

if $x\left[i\right]=U$ then |

v.children.append(node) |

stack.push(u) |

$u\leftarrow v$ |

else if $x\left[i\right]=I$ then |

v.children.append(node) |

else if $x\left[i\right]=H$ then |

u.children.append(node) |

else if $x\left[i\right]=D$ then |

u.children.append(node) |

$u\leftarrow $ stack.pop() |

$v\leftarrow $ node |

return root |

#### 3.2. Mixing Time Results

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

- With probability $1-\frac{m-2k}{4m}$, set $({X}_{t+1},{Y}_{t+1})=({X}_{t},{Y}_{t})$.
- Otherwise, pick a random index $j\in [m-2k]$. Let $a\in \{H,I\}$ be a random symbol such that $Pr(a=H)=\frac{{e}^{-\alpha}}{{e}^{-\alpha}+{e}^{-\beta}}$ and $Pr(a=I)=\frac{{e}^{-\beta}}{{e}^{-\alpha}+{e}^{-\beta}}$. Now let ${X}_{t+1}$ and ${Y}_{t+1}$ be ${X}_{t}$ and ${Y}_{t}$ respectively, each with the jth symbol changed to a.

**Lemma**

**10.**

**Proof.**

**Lemma**

**11.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 4. Discussion and Conclusions

#### 4.1. Applications to RNA Modeling

#### 4.2. Possibility of a Dynamic Programming Approach

#### 4.3. Possibility of an SCFG Approach

#### 4.4. Extended Applications

#### 4.5. Independent Mathematical Research Interests

#### 4.6. Limitations and Future Directions

- Can the mixing time bound in our main result be improved?
- Is there a rapidly mixing chain, with the same stationary distribution studied here, whose transitions correspond naturally to moves on the set plane trees? Mixing time bounds on the chain of matching exchange moves, as defined in [63], would be especially interesting, as such a chain may relate to RNA folding kinetics.
- Is there a rapidly mixing chain converging to the Gibbs distribution using the full energy function for the utilized NNTM model [16]? The chain presented here uses only the parameters $\alpha $ and $\beta $, setting $\gamma =0$.
- Is there a stochastic context-free grammar which generates secondary structures (in our simplified model or using the full NNTM) according to a Gibbs distribution with NNTM energy?

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RNA | ribonucleic acid |

NNTM | Nearest Neighbor Thermodynamic Model |

SCFG | stochastic context free grammar |

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**Figure 1.**A ribonucleic acid (RNA) secondary structure for one of the combinatorial RNA sequences used in this work and its corresponding plane tree. The ordering of the edges in the plane tree is derived from the 3’ to 5’ ordering of the RNA sequence. Note that the exterior loop corresponds to the root of the plane tree. The diagram in (

**a**) was generated by ViennaRNA [19]. (

**a**) A maximally-paired secondary structure for A

^{4}(C

^{5}GA

^{4}CG

^{5}A

^{4})

^{4}has 4 helices; (

**b**) The corresponding plane tree has 4 edges and encodes the branching pattern seen in the secondary structure.

**Figure 2.**A plane tree with edges labeled according to the bijection $\mathsf{\Phi}$, along with its corresponding 2-Motzkin path.

**Figure 3.**The four level decomposition of ${\mathfrak{M}}_{m}^{2}$ (

**left**), and the projection chains corresponding to each decomposition (

**right**).

**Table 1.**Nearest Neighbor Thermodynamic Model (NNTM) parameters and resulting energy functions. Energy functions are of the form $\alpha {d}_{0}+\beta {d}_{1}+\gamma r$.

Y | Z | Turner | a | b | c | h | f | i | g | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

C | G | 89 | 4.6 | 0.4 | 0.1 | $-10.9$ | 3.8 | 3.0 | $-1.6$ | $-0.9$ | $-1.8$ | $-1.7$ |

G | C | 89 | 4.6 | 0.4 | 0.1 | $-16.5$ | 3.5 | 3.0 | $-1.9$ | $-0.9$ | $-1.2$ | $-1.7$ |

C | G | 99 | 3.4 | 0 | 0.4 | $-12.9$ | 4.5 | 2.3 | $-1.6$ | 2.3 | 1.3 | $-0.4$ |

G | C | 99 | 3.4 | 0 | 0.4 | $-16.9$ | 4.1 | 2.3 | $-1.9$ | 2.2 | 1.9 | $-0.4$ |

C | G | 04 | 9.3 | 0 | $-0.9$ | $-12.9$ | 4.5 | 2.3 | $-1.1$ | $-2.8$ | $-3.0$ | 0.9 |

G | C | 04 | 9.3 | 0 | $-0.9$ | $-16.9$ | 4.1 | 2.3 | $-1.5$ | $-2.8$ | $-2.2$ | 0.9 |

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## Share and Cite

**MDPI and ACS Style**

Kirkpatrick, A.; Patton, K.; Tetali, P.; Mitchell, C.
Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications. *Math. Comput. Appl.* **2020**, *25*, 67.
https://doi.org/10.3390/mca25040067

**AMA Style**

Kirkpatrick A, Patton K, Tetali P, Mitchell C.
Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications. *Mathematical and Computational Applications*. 2020; 25(4):67.
https://doi.org/10.3390/mca25040067

**Chicago/Turabian Style**

Kirkpatrick, Anna, Kalen Patton, Prasad Tetali, and Cassie Mitchell.
2020. "Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications" *Mathematical and Computational Applications* 25, no. 4: 67.
https://doi.org/10.3390/mca25040067