# 40Multibranch Loops

## 40.1 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]

## 40.2 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

## 40.3 Parameter Tables

Tables of parameters are available in html format.