# 19Multibranch Loops

## 19.1 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 each nucleotide or helix end can participate in only one of these favorable interactions.

When there are six or fewer unpaired nucleotides in the loop, initiation is predicted using:

ΔG°37 initiation = a + b×[number of unpaired nucleotides] + c×[number of branching helices]

where a, b, and c are parameters.

For multibranch loops with greater than six unpaired nucleotides, a logarithmic dependence on the number of upaired nucleotides is used:

ΔG°37 initiation = a + 6b + (1.1 kcal/mol)×ln([number of unpaired nucleotides]/6) + c×[number of branching helices]

## 19.2 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 + 6b + (1.1 kcal/mol)×ln([number of unpaired nucleotides]/6) + c×[number of branching helices]

ΔG°37 initiation = 10.1 kcal/mol + 6×(–0.3 kcal/mol) + (1.1 kcal/mol)×ln(8/6) + 4×(–0.3 kcal/mol)

ΔG°37 initiation = 7.4 kcal/mol

### 19.2.1 Total Folding Free Energy Change

ΔG°37 multibranch loop = ΔG°37 initiation + ΔG°37 stacking = 7.4 kcal/mol – 5.9 kcal/mol = 1.5 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.

## 19.3 Tables

Tables of parameters are available in html format.

## 19.4 References

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

Mathews, D.H., Sabina, J., Zuker, M. and Turner, D.H. (1999) Expanded sequence dependence of thermodynamic parameters provides improved prediction of RNA secondary structure. J. Mol. Biol., 288, 911-940.