# 9Multibranch Loops

## 9.1 Folding Free Energy Change

Multibranch loops stabilities are predicted using the following equation:

ΔG°37 multibranch = ΔG°37 initiation + ΔG°37 stacking

where the stacking is the optimal configuration of dangling ends, terminal mismatches, or coaxial stacks, noting that a nucleotide or helix end can participate in only one of these favorable interactions.

Initiation is predicted using:

ΔG°37 initiation = a + b×[average asymmetry] + c×[number of branching helices] + ΔG°37 strain(three-way branching loops with fewer than two unpaired nucleotides)

where the average asymmetry is calculated as:

average asymmetry = min[2.0, mean difference in unpaired nucleotides on each side of each helix]

## 9.2 Folding Enthalpy Change

Similar to free energy change, multibranch loops enthalpy changes are predicted using the following equation:

ΔH°multibranch = ΔH°initiation + ΔH°stacking

where the stacking is the configuration of dangling ends, terminal mismatches, or coaxial stacks with lowest folding free energy change.

Initiation is predicted using an equation analagous to that folding free energy initiation:

ΔH°initiation = a + b×[average asymmetry] + c×[number of branching helices] + ΔH°strain(three-way branching loops with fewer than two unpaired nucleotides)

where the average asymmetry is calculated as:

average asymmetry = min[2.0, mean difference in unpaired nucleotides on each side of each helix]

## 9.3 Example

### Free Energy Change

#### Prediction of Stacking

The predicted stacking configuration is the one with lowest free energy change. There are eight relevant configurations.

#### Configuration 1:

Helix 1 with $$3'$$ dangling U, Helix 2 with terminal mismatch, Helix 3 with $$3'$$ dangling A, and Helix 4 with $$5'$$ dangling C

ΔG°37 = ΔG°37(UA with $$3'$$ dangling U) + ΔG°37(CG followed by GA) + ΔG°37(GC with $$3'$$ dangling A) + ΔG°37(GC with $$5'$$ dangling C)

ΔG°37 = –0.1 kcal/mol – 1.4 kcal/mol – 1.1 kcal/mol – 0.3 kcal/mol

ΔG°37 = –2.9 kcal/mol

#### Configuration 2:

Helix 1 with $$3'$$ dangling U, Helix 2 with $$5'$$ dangling A, Helix 3 with terminal mismatch, and Helix 4 with $$5'$$ dangling C

ΔG°37 = ΔG°37(UA with $$3'$$ dangling U) + ΔG°37(CG with $$5'$$ dangling A) + ΔG°37(GC followed by AG) + ΔG°37(GC with $$5'$$ dangling C)

ΔG°37 = –0.1 kcal/mol – 0.2 kcal/mol – 1.3 kcal/mol – 0.3 kcal/mol

ΔG°37 = –1.9 kcal/mol

#### Configuration 3:

Helix 1 in flush coaxial stack with helix 4, Helix 2 with terminal mismatch, and Helix 3 with $$3'$$ dangling A

ΔG°37 = ΔG°37(GC followed by AU) + ΔG°37(CG followed by GA) + ΔG°37(GC with $$3'$$ dangling A)

ΔG°37 = –2.35 kcal/mol – 1.4 kcal/mol – 1.1 kcal/mol

ΔG°37 = –4.9 kcal/mol

#### Configuration 4:

Helix 1 in flush coaxial stack with helix 4, Helix 2 with $$5'$$ dangling A, and Helix 3 with terminal mismatch

ΔG°37 = ΔG°37(GC followed by AU) + ΔG°37(CG with $$5'$$ dangling A) + ΔG°37(GC followed by AG)

ΔG°37 = –2.35 kcal/mol – 0.2 kcal/mol – 1.3 kcal/mol

ΔG°37 = –3.9 kcal/mol

#### Configuration 5:

Helix 1 with $$3'$$ dangling U, Helix 2 in mismatch–mediated coaxial stack with helix 3 with GA intervening mismatch, and Helix 4 with $$5'$$ dangling C

ΔG°37 = ΔG°37(UA with $$3'$$ dangling U) + ΔG°37(CG followed by GA) + ΔG°37(Discontinuous Backbone Stack) + ΔG°37(GC with $$5'$$ dangling C)

ΔG°37 = –0.1 kcal/mol – 1.4 kcal/mol – 2.1 kcal/mol – 0.3 kcal/mol

ΔG°37 = –3.9 kcal/mol

#### Configuration 6:

Helix 1 with $$3'$$ dangling U, Helix 2 in mismatch–mediated coaxial stack with helix 3 with AG intervening mismatch, and Helix 4 with $$5'$$ dangling C

ΔG°37 = ΔG°37(UA with $$3'$$ dangling U) + ΔG°37(Discontinuous Backbone Stack) + ΔG°37(GC followed by AG) + ΔG°37(GC with $$5'$$ dangling C)

ΔG°37 = –0.1 kcal/mol – 2.1 kcal/mol – 1.3 kcal/mol – 0.3 kcal/mol

ΔG°37 = –3.8 kcal/mol

#### Configuration 7:

Helix 1 in flush coaxial stack with helix 4 and Helix 2 in mismatch–mediated coaxial stack with helix 3 with GA intervening mismatch

ΔG°37 = ΔG°37(GC followed by AU) + ΔG°37(CG followed by GA) + ΔG°37(Discontinuous Backbone Stack)

ΔG°37 = –2.35 kcal/mol – 1.4 kcal/mol – 2.1 kcal/mol

ΔG°37 = –5.9 kcal/mol

#### Configuration 8:

Helix 1 in flush coaxial stack with helix 4 and Helix 2 in mismatch–mediated coaxial stack with helix 3 with AG intervening mismatch

ΔG°37 = ΔG°37(GC followed by AU) + ΔG°37(Discontinuous Backbone Stack) + ΔG°37(GC followed by AG)

ΔG°37 = –2.35 kcal/mol – 2.1 kcal/mol – 1.3 kcal/mol

ΔG°37 = –5.8 kcal/mol

Configuration 7 has the lowest folding free energy change of –5.9 kcal/mol.

#### Initiation Free Energy Change

ΔG°37 initiation = a + b×[average asymmetry] + c×[number of branching helices] + ΔG°37 strain(three–way branching loops with fewer than two unpaired nucleotides)

ΔG°37 initiation = 9.25 kcal/mol + (0.91 kcal/mol)×[average asymmetry] + (–0.63 kcal/mol)×[4]

Average asymmetry = min[2.0,(2+1+4+5)/4] = min[2.0,3.0] = 2.0

ΔG°37 initiation = 9.25 kcal/mol + (0.91 kcal/mol)×[2] + (–0.63 kcal/mol)×[4]

ΔG°37 initiation = 8.6 kcal/mol

#### Total Folding Free Energy Change

ΔG°37 multibranch loop = ΔG°37 initiation + ΔG°37 stacking = 8.6 kcal/mol – 5.9 kcal/mol = 2.7 kcal/mol

### 9.3.1 Enthalpy Change

#### Prediction of Stacking

The stacking configuration is fixed by the prediction of folding free energy change and is configuration 7 above.

ΔH°= ΔH°(GC followed by AU) + ΔH°(CG followed by GA) + ΔH°(Discontinuous Backbone Stack)

ΔH° = –12.44 kcal/mol – 8.2 kcal/mol – 8.46 kcal/mol

ΔH° = –29.1 kcal/mol

#### Initiation Enthalpy Change

ΔH°initiation = a + b×[average asymmetry] + c×[number of branching helices] + ΔH°strain(three–way branching loops with fewer than two unpaired nucleotides)

ΔH°initiation = 38.9 kcal/mol + (12.9 kcal/mol)×[average asymmetry] + (–11.9 kcal/mol)×[4]

Average asymmetry = min[2.0,(3+2+4+5)/4] = min[2.0,3.0] = 2.0

ΔH°initiation = 38.9 kcal/mol + (12.9 kcal/mol)×[2] + (–11.9 kcal/mol)×[4]

ΔH°initiation = 17.1 kcal/mol

#### Total Folding Enthalpy Energy Change

ΔH°multibranch loop = ΔH°initiation + ΔH°stacking = 17.1 kcal/mol – 29.1 kcal/mol = –12.0 kcal/mol

Note that helices 1 and 2 are separated by two unpaired nucleotides and cannot stack coaxially. Similarly, helices 3 and 4 are too distant to stack coaxially. Also note that coaxial stacking is only allowed between adjacent helices and hence, for example, helices 1 and 3 cannot stack coaxially.

## 9.4 Parameter Tables

Tables of parameters are available in html format.

## 9.5 References

The multibranch loop nearest neighbor parameters for initiation free energy change were reported in:

Mathews, D.H. and Turner, D.H. (2002) Experimentally derived nearest neighbor parameters for the stability of RNA three- and four-way multibranch loops. Biochemistry, 41, 869-880.

The enthalpy change parameters were reported in:

Lu, Z.J., Turner, D.H. and Mathews, D.H. (2006) A set of nearest neighbor parameters for predicting the enthalpy change of RNA secondary structure formation. Nucleic Acids Res., 34 4912 - 4924.

The experimental data for the fit of the parameters were taken from:

1. Diamond, J.M., Turner, D.H. and Mathews, D.H. (2001) Thermodynamics of three-way multibranch loops in RNA. Biochemistry, 40, 6971-6981.
2. Mathews, D.H. and Turner, D.H. (2002) Experimentally derived nearest neighbor parameters for the stability of RNA three- and four-way multibranch loops. Biochemistry, 41, 869-880.