Robustness and evolvability are highly intertwined properties of biological systems. The relationship between these properties determines how biological systems are able to withstand mutations and show variation in response to them. Computational studies have explored the relationship between these two properties using neutral networks of RNA sequences (genotype) and their secondary structures (phenotype) as a model system. However, these studies have assumed every mutation to a sequence to be equally likely; the differences in the likelihood of the occurrence of various mutations, and the consequence of probabilistic nature of the mutations in such a system have previously been ignored. Associating probabilities to mutations essentially results in the weighting of genotype space. We here perform a comparative analysis of weighted and unweighted neutral networks of RNA sequences, and subsequently explore the relationship between robustness and evolvability.

1. Karthik Raman
Department of Biotechnology
Bhupat and Jyoti Mehta School of
Biosciences
REVISITING ROBUSTNESS AND
EVOLVABILITY: EVOLUTION ON
WEIGHTED GENOTYPE
NETWORKS

2. INTRODUCTION

3. What is Robustness?
 Ability to continue normal function in the face
of perturbations / resist change
 Defining features of many biological
systems/networks
 Many biological networks can show the same
function – their function is robust to variations

4. What is Evolvability?
 “The ability to produce phenotypic
diversity, novel solutions to the problems
faced by organisms and evolutionary
innovations”a
 Ability to change
 Again, a common feature of biological
…
aWagner A (2008) Bioessays, 30: 367–373

5. How can robustness and
evolvability co-exist?

6. NEUTRAL NETWORKS

7. Neutral Networks
 Genotype space
 Nodes: Genotypes
 Edges: mutation/simple
change
 Neutral Networks: Genotypes sharing the same
phenotype

8. Neutral Networks
 Genotype space
 Nodes: Genotypes
 Edges: mutation/simple
change
 Genotypes mapped to
phenotypes
 Neutral Networks: Genotypes sharing the same
phenotype

9. Neutral Networks
 Genotype space
 Nodes: Genotypes
 Edges: mutation/simple
change
 Genotypes mapped to
phenotypes
 Largest Neutral Network
 Neutral Networks: Genotypes sharing the same
phenotype

10. Neutral Networks
 Genotype space
 Nodes: Genotypes
 Edges: mutation/simple
change
 Genotypes mapped to
phenotypes
 Largest Neutral Network
 Smaller Neutral Network
 Neutral Networks: Genotypes sharing the same
phenotype

11. Neutral Networks
 Genotype space
 Nodes: Genotypes
 Edges: mutation/simple
change
 Genotypes mapped to
phenotypes
 Largest Neutral Network
 Smaller Neutral Network
 Smaller and more
neutral network
 Neutral Networks: Genotypes sharing the same
phenotype

12. Neutral Networks
 Genotype space
 Nodes: Genotypes
 Edges: mutation/simple
change
 Genotypes mapped to
phenotypes
 Largest Neutral Network
 Smaller Neutral Network
 Smaller and more
neutral network
 Multiple neutral sets
 Neutral Networks: Genotypes sharing the same
phenotype

13. Neutral Networks
 Genotype space
 Nodes: Genotypes
 Edges: mutation/simple
change
 Genotypes mapped to
phenotypes
 Largest Neutral Network
 Smaller Neutral Network
 Smaller and more
neutral network
 Multiple neutral sets
 Neutral Networks: Genotypes sharing the same
phenotype
 The genotype space is covered with multiple neutral
networks

14. Neutral Networks vs. Robustness

15. Neutral Networks vs. Robustness

16. Neutral Networks vs. Robustness
 Low robustness: more deleterious mutations
 High robustness: likely to encounter more novel phenotypes
Wagner A (2008) Nat Rev Genet 9:965–
974

17. Robustness vs. Evolvability
 Robustness and evolvability — both correlate with neutral network
size/connectivity
 Robust phenotypes tend to have higher evolvability: Populations evolving on
evolving on larger neutral networks have greater access to variation*
Ciliberti S, Martin OC & Wagner A (2008) PNAS 104:13591-6

18. Robustness vs. Evolvability
 Definition of mutation depends on the level of organisation
 Genotype: sequence
 Phenotype: structure
 Mutation to a Genotype
 Phenotype does not change: Neutral mutation
 Phenotype changes: Non-neutral mutation
 Robustness: Ability of systems to withstand mutations
 Evolvability: Ability of mutations to produce heritable
phenotypic variation
 If a system is highly robust to mutations then mutations cannot
lead to variation, which means less evolvability!

19. Resolving the Paradox
 Seminal paper by Andreas Wagner
 Uses RNA as model system
 Builds on original work by
Schuster, P., Fontana, W., Stadler, P. & Hofacker, I. (1994) “From
sequences to shapes and back—a case-study in RNA secondary
structures” Proc. R. Soc. B 255:279–284

20. RNA GENOTYPE SPACE

21. Neutral Networks of RNA sequences
 Genotype: RNA sequence (length L)
 Phenotype: RNA secondary structure
 Neutral neighbours
 1-neighbourhood of genotype:
set of neighbours = 3L sequences
 1-neighbourhood of phenotype:
set of sequences that differ from
sequences that fold into the structure by exactly one
nucleotide
 High genotype robustness ⇒ low genotype evolvability
 High phenotype robustness ⇒ high phenotype
evolvability

22. Weighting the Genotype Space
 Major assumption in all previous studies (multiple systems) is
that every mutation is equally likely
 Every edge corresponds to a mutation in this system: single
nucleotide change, which is either transition or transversion
 Relative rates of occurrence are given by the transition–
transversion ratio, kappa (κ): 2.1–2.5 across genomesa
 Depending on the type of mutation, each mutation can be
associated with a probability of occurrence
aDePristo MA et al (2011) Nature Genetics 43:491–8

23. Definitions
Property Old Definitiona New Definition
Genotype
Robustness
Number of neutral
neighbours of G
Probability of reaching a neutral
neighbour of G
Genotype
Evolvability
Number of different
structures in N1 of G
Summation of the mean
probabilities of evolving to a
structure different from P, in N1
of G
Phenotype
Robustness
Mean genotype robustness
of all G’s with P
Mean genotype robustness of
all G’s with P
Phenotype
Evolvability
Number of different
structures in N1 of P
Mean probability of evolving
from P to a different structure
summed over all different
structures in N1 of P
 Genotype: G, Phenotype: P, 1-neighbourhood:
N1
 Inverse folding sequences for a structure
aWagner A (2008) Proc Biol Sci 275:91–100

24. Weighting the network
changes robustness and
evolvability …
Results

25. High genotype robustness corresponds to low genotype evolvability
κ Mean genotype robustness Mean genotype evolvability
0.5 0.42 0.28
2.5 0.48 0.25
10 0.50 0.24

26. High phenotype robustness corresponds to high phenotype evolvability
κ Mean phenotype robustness Mean phenotype evolvability (*10-4)
0.5 0.32 4.40
2.5 0.39 3.96
10 0.42 3.73

27. Population evolution: Nµ = 100

28. Population evolution at Nµ = 100 for 103 structures
p-values of pair-wise Wilcoxon signed rank test for all three pairs
of datasets were <10-17. Correlation values are Spearman’s r values
with p-values less than 10-17 for κ = 0.5 and 2.5, and less than 10-4
for κ = 10.
 Cumulative novel phenotypes encountered at the end of 10
generations of mutations
 A decrease in value is observed upon weighting the genotype
space
κ Cumulative novel
phenotypes
Correlation with structure
frequency
0.5 1089±301 0.22
2.5 1013±253 0.16
10 830±206 0.12

29. Population evolution: Nµ = 1

30. Population evolution at Nµ = 1 for 103 structures
p-values of pair-wise Wilcoxon signed rank test for the
datasets was <10-3. Correlation values are Spearman’s r values
with all p-values<10-4
 Cumulative novel phenotypes encountered at the end of
100 generations of mutations
 A modest but significant decrease in value is observed
upon weighting the genotype space
κ Cumulative novel
phenotypes
Correlation with structure
frequency
0.5 244±63 0.22
2.5 240±58 0.13

31. Summary
 Ignoring mutational probabilities
underestimates robustness and overestimates
evolvability
 Incorporating weighting does not substantially
affect the nature of relationship

32. BASE COMPOSITION BIAS

33. Base composition bias
 GC content varies across genomesa
 GC pairs are more stable than AU pairs
 AU-rich (>80% AU) RNA sequences are less thermally
stable
 AU-rich RNA structures relatively sparser
 Higher fraction of AU-rich sequences (compared to unbiased
sequences) do not fold into a stable secondary structure
 Do these factors affect robustness and evolvability in AU-
rich sequence space?
aBirdsell JA (2002) Molecular Biology and Evolution 19:1181–1197

34. Minimum Free Energy (MFE) distribution
Structures formed by1 million AU-rich and normal sequences Structures common to both
sequence spaces

35. Genotype robustness and evolvability
κ Mean genotype robustness Mean genotype evolvability
Normal AU-rich Normal AU-rich
0.5 0.42 0.48 0.28 0.18
2.5 0.48 0.56 0.25 0.16
10 0.50 0.60 0.24 0.15

36. Phenotype robustness and evolvability
 Differences in these properties in AU-rich vs normal
space, for neutral networks of the same phenotype?
 Inversely fold AU-rich sequences for a given
phenotype
 Inversely fold normal sequences
 Structure preserving random-walk starting from
these sequences towards AU-rich space
 Resulting population of AU-rich sequences can be
used for further computations

37. Phenotype robustness and evolvability
κ Mean phenotype robustness Mean phenotype evolvability (*10-4)
Normal AU-rich Normal AU-rich
0.5 0.32 0.37 4.40 1.84
2.5 0.39 0.44 3.96 1.76
10 0.43 0.48 3.73 1.72

39. Population evolution at Nµ = 100 for 103 structures
p-values of pair-wise Wilcoxon signed rank test for all 3 pairs of
datasets were <10-17. Correlation values are Spearman’s r values
with all p-values <10-17.
 Cumulative novel phenotypes encountered at the end of 10
generations of mutations: AU-rich populations access less
variation compared to normal populations
κ Cumulative novel
phenotypes
Correlation with structure
frequency
Normal AU-rich Normal AU-rich
0.5 1089±301 778±214 0.22 0.25
2.5 1013±253 774±189 0.16 0.21
10 830±206 652±170 0.12 0.20

40. Thank You

41. Population evolution at Nµ = 1 for 103 structures
p-values of pair-wise Wilcoxon signed rank test for all 3 pairs of
datasets were <10-17. Correlation values are Spearman’s r values
with all p-values <10-4.
 Cumulative novel phenotypes encountered at the end of 100
generations of mutations
 AU-rich populations access less variation compared to normal
populations
κ Cumulative novel
phenotypes
Correlation with structure
frequency
Normal AU-rich Normal AU-rich
0.5 244±63 175±49 0.22 0.18
2.5 240±58 181±46 0.13 0.15

42. Summary
 AU-rich genotypes are more robust and less evolvable
 Neutral networks of phenotypes have higher robustness
and lesser evolvability in AU-rich space compared to
normal space
 AU-rich populations evolving on a phenotype’s neutral
network access less variation than normal populations
 Results indicative of the restrictive nature of AU-rich space
– lesser accessibility to variation

43. Acknowledgements
Andreas WagnerRaghavendran Partha
Funding

44. More Information
datahub.io/dataset/weighted-genotype-networks-of-rna
(2014) PLoS ONE 9:e112792 PMID: 25390641

45. Thank You
Questions?

47. AU rich sequences are ~unique