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- - - - tMRCA 2-deme symmetric ts = same deme, td = different demes
      - Could be used to infer migration rates and effective population sizes, but requires tree+migration history (labeled edge tree) so not used in practice but sometimes with bayesian methods
  - - - Basic parameters of a simple birth-death model that accounts for sampling the population dynamical process
        
        We could get values for birth and death parameters from a regression of this plot, choosing intervals and accounting for variances is too problematic
        
        Population of I at time t (deterministic eq.)
        
        To get a ML for the parameters beta and delta we need to derive:
        
        To do that we need to derive:
        
        The probability of extinction after time t with parameters beta and delta and the probability of their being 1 survivor after time t
        
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    - - Kingsmans Coalescent can be derived from this model
        
        Often used to infer population sizes and size dynamics
        
        We can estimate the probability of a phylogenetic tree by using the probability of k linages coalescing in time delta t. we do this by taking the product of probabilities at between each coalescent events i.e. the internal nodes
        
        We can then infer population size by maximizing the likelihood of N
        
        We can also make N a function of t N(t), and build models that infer a dynamic population size and compute likelihoods
        
        Classic WF does not account for the structure of the population, that part of the reason why we called the inferred population size from a Kings coalescent the Effective population size
        
        The Coalescent can also be derived as limit of many other population processes which is why it is sometimes called robust as all heck
      - 2 linages coalescing
        
        2 in k lineages coalescing
        
        2NG is upper bound on tree
- - - - Parameterized scoring function for match, mismatch and gaps
        
        Smith-Waterman (Local)
        
        Needleman-Wunch (Global)
        
        edges are negative gaps, start in bottom, allow negative vals
        
        Exhaustive search (Non-DP)
    - - Easy to identify features such as indels, repeats and inversions
    - - BLAST
        
        Break sequence into k-mers and look for k-mers in database, allow mismatches but use scoring function, expand alignment around k-mer matches iteratively, keep alignments above score threshold
        
        50-100 times faster, produces non optimal alignments
  - - - Works but is slow
    - - Progressive Alignments
      - Select reference strain and pairwise align to it
- - - - ODDS RATIO = OR
        Evaluate alleles healthy vs. diseased
        
        Statistical test to get p-values
        From contingency table
        
        Fisher's Exact Test
        
        Works for 2-classes (Diseased and healthy)
        Null hypothesis, we assume classes don't matter, chance of being either follows hypergeometric distribution
        
        Only works for small numbers
        
        Pearson's Chi^2
        
        Works for big numbers
        
        P-value is the probability of observing a a certain value x or more extreme under the assumption of some null hypothesis
        
        Significance (alpha) level says how ridiculous our observation has to be to just say what the hell and reject the null hypothesis,
- - - - Vanilla
    - - Has special parameter for transitions ( T-C and A-G)
    - - Same as TN93 but there is only one transition parameters instead of 2
    - - Has symmetric parameters for every transition and transversion (most general time reversible model)
    - - Is the most general, separate non symmetric parameter for every rate, not time reversible and hard to use
    - - Includes equilibrium frequencies and has a separate parameter for each type of transition
    - - We say the gamma dist has one parameter = a so a = b and the mean is 1
  - - - Transition probability of event occurring at at some rate alpha
        
        Cumulative density function
    - - Series of random experiments through time
        
        Lives on state space S, where we jump between states of S
        
        Memorylessness (given some state space, the probability of transition to different space is only effected by your current sate and not the state you were in before
        
        Continuous or discrete
        
        Time homogenous if transition probabilities don't change over time
  - - - JC69 + gamma
        
        p=x/n
- - - - Pairwise distances/Overall similarity of sequences
        
        Search for optimality
        
        least-squares methods
        
        Combinatorial explosion means we have to check all trees
        
        Algorithmic methods
        
        UPGMA
        
        Distance matrix in, Ultrameric Tree out
        
        If there is an ultra meric tree that induces the distance matrix, then UPGMA finds it (The tree distance matrix and the original distance matrix may not always match)
        
        Assumes
        1.All sequences sampled at same time
        2.Constant evolution rate over time
        
        Neighbour-Joining
        
        Neighbor-joining relaxes assumptions
        
        Produces unrooted tree
      - Only pairwise distances, no higher order correlations between seqeunces
    - - Based on shard characteristics
        
        Parsimony Method
        
        Parsimony score is the minimum number of mutations needed to explain some tree
        
        Fitch's Algorithm efficiently finds the parsimony score of a tree
        
        We have to visit each internal node (n-1) and for each node we must go over each character in sequence so complexity is m(n-1)
        
        Parsimony tree is the tree, given some sequences, has the lowest parsimony score.
        
        Rooting an unrooted tree will not alter the parsimony score
        
        NP-Complete because all trees must be evaluated
        
        Parsimony is not statistically consistent due long branch attraction in the felsenstein zone
        
        Cladistic inference is in practice statistically inconsistent, does not consider back substitutions which leads to the felsenstein zone problem, It also slow, sometimes used for macroevolution by cladists
    - - Based on evolutionary models, character based
        
        Maximum likelihood
        
        Take the product of proximities over all locations over all internal nodes for each nucleotide
        
        Felsenstein Pruning Algorithm
        
        Time reversibility implies that differently rooted trees should have the same likelihood
        
        Bayesian Inference
    - - How many unrooted trees with n tips
      - How many rooted trees with n tips
  - - - The nearest-neighbour interchange (NNI) algorithm
        Simply pick an internal edge connecting subtrees and swap it with another
      - Branch swapping by subtree pruning and regrafting
        (SPR)
        Basically remove any subtree and stick it somewhere else
      - Branch swapping by tree bisection and reconnection (TBR)
        Break tree at some edge and the reattach at a different edge (sounds the same as NNI)
    - - Likelihood Ratio testing
        
        Can be used to compare different models but requires that that models are nested within each other, for example the JC69 model is nested within K80 or how K80 is nested within UNREST
- - - - Rather, calculate the if changes in traits are correlated on branches
        
        Discrete Characters
        
        For each branch, look if the traits changes together, then use fishers test
        
        This is hard because we don't have a perfect history of traits along branches, also as changes are more likely to happen on long branches we might want to control for the branch lengths
        
        Continuous Characters
        
        Linear regression is analog to fishers test
        
        Must account for phylogenies
        
        Model character traits with Brownian motion model,a time-continuous stochastic process on a continuous state space
        
        Memorylessness
        
        Variance scales with branch length analog to transition probabilities scale with time
        
        To account for this we can use the contrast method
        
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        Linear regression assumes independent samples that have the same normally distributed errors
        
        Assumption broken for sampling of related individuals
        
        Furthermore as the variance scales with branch lengths, the assumption about normally distributed erros is also broken