MODELING INTER-MOTIF DEPENDENCE WITHOUT INCREASING THE COMPLEXITYZhizhuo Zhang

PWM MODEL

1 2 3 4 5 6

A 0.1 0 0 1 0.4 0.4

C 0 0.1 0 0 0.4 0

G 0.1 0 1 0 0.2 0.2

T 0.8 0.9 0 0 0 0.4

PositionalWeightMatrix (PWM)

| | 4

1 1

log(P(x| ))= ( ( ), ) log( )

where ( , ) is indicator function

x

iji j

x i j

TTGACTTCGACTTTGACTTTGAAAATGAGGTTGAAAGTGAAATTGACTTTGAGGTTGAAA

HIGH -ORDER DEPENDENCY

1st -order

2mer P4-5

CT 0.4

AA 0.4

GG 0.2

CC 0

AC 0

…. 0

TT 0

TTGACTTCGACTTTGACTTTGAAAATGAGGTTGAAAGTGAAATTGACTTTGAGGTTGAAA

HIGH -ORDER DEPENDENCY

Assume only one dependency group

1...| |

4

1, 1 2

log(P(x| , ))= ( ( ), ) log( ) ( , ) log( ( ( )))

where ( , ) is indicator function

xij

i i j I

x i j I i x I

TWO MODELING PRINCIPLES

Inter-dependence bases only exists in the diverged positions.

There is no inter-dependence relationship across the conserved base.

PRINCIPLE ONE

People use KL-Divergence to measure the dissimilarity between two probability distribution

To show the KL-divergence between K+1 order distribution and K order distribution + 0 order distribution is small when the K+1 position base is very conserved.

PRINCIPLE ONEThe KL-divergence between K+1 order distribution and K order distribution + 0 order distribution is followed:

= 𝑃1:𝑘+1(𝐴,𝑥)𝑙𝑜𝑔 𝑃1:𝑘+1ሺ𝐴,𝑥ሻ𝑃1:𝑘ሺ𝐴ሻ𝑃𝑘+1ሺ𝑥ሻ𝐴=ሾ𝑎𝑐𝑔𝑡ሿ𝑘,𝑥=ሾ𝑎𝑐𝑔𝑡ሿ

= 𝑃1:𝑘+1(𝐴,𝑥)𝑙𝑜𝑔𝑃1:𝑘+1ሺ𝐴,𝑥ሻ𝑃1:𝑘ሺ𝐴ሻ𝐴=ሾ𝑎𝑐𝑔𝑡ሿ𝑘,𝑥=ሾ𝑎𝑐𝑔𝑡ሿ − 𝑃1:𝑘+1(𝐴,𝑥)𝑙𝑜𝑔𝑃𝑘+1ሺ𝑥ሻ𝐴=ሾ𝑎𝑐𝑔𝑡ሿ𝑘,𝑥=ሾ𝑎𝑐𝑔𝑡ሿ

= ቌ 𝑃1:𝑘+1(𝐴,𝑥)𝑙𝑜𝑔𝑃1:𝑘+1ሺ𝐴,𝑥ሻ𝑃1:𝑘ሺ𝐴ሻ𝐴=ሾ𝑎𝑐𝑔𝑡ሿ𝑘,𝑥=ሾ𝑎𝑐𝑔𝑡ሿ ቍ+ 𝐻𝑘+1ሺ𝑋ሻ = 𝐻1:𝑘ሺ𝐴ሻ− 𝐻1:𝑘+1ሺ𝐴,𝑋ሻ+ 𝐻𝑘+1ሺ𝑋ሻ = −𝐻ሺ𝑥ȁ𝐴ሻ+ 𝐻𝑘+1ሺ𝑋ሻ ≤ 𝐻𝑘+1(𝑋)

PRINCIPLE TWOCys2His2 Zinc Finger DNA-binding family, which is the largest known DNA-binding family in multi-cellular organisms.

Independent

CONTROL THE COMPLEXITY

The larger the dependence group, the more parameters, the easier to overfit.

We want to model the k-order dependence using the same number of parameters as (k+1) independent position PWM. (i.e.,4k+4 parameters)

CONTROL THE COMPLEXITY

1 2 3 4 5 6

A 0.1 0 0 1 CT=0.4

AC=0

C 0 0.1 0 0 AA=0.4

CA=0

G 0.1 0 1 0 GG=0.2

TT=0

T 0.8 0.9 0 0 CC=0 Other=0

TTGACTTCGACTTTGACTTTGAAAATGAGGTTGAAAGTGAAATTGACTTTGAGGTTGAAA

Dependence PositionalWeightMatrix (PWM)

CONTROL THE COMPLEXITY

Model the problem: Given a set of binding site sequences X (each is

length k), find a DPWM Ω maximize the likelihood P(X| Ω) (or minimize the KL-divergence), with 4k parameters

We can prove that taking top 4k-1 kmer probability as the first 4k-1 paramter value is the best solution:

Let the alphabet index for 𝐴= ሾ𝑎𝑐𝑔𝑡ሿ𝑘 is 1,2,3,…, 4𝑘, the k-order dependency model is built according to the following rules:

𝑃𝑘ሺ𝑖ሻ= ቌ

𝑃𝑡ሺ𝑖ሻ 𝑖 ∈ሾ1,4𝑘− 1ሿ𝑚𝑘 𝑖 ∈ሾ4𝑘,4𝑘ሿ,𝑚𝑘 = 𝑃𝑡ሺ𝑖ሻ4𝑘 − 4𝑘+ 14𝑘

𝑖=4𝑘ቍ

𝑓𝑖𝑛𝑑 𝑷𝒌 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑃𝑡(𝐴)𝑙𝑜𝑔𝑃𝑘ሺ𝐴ሻ𝐴=ሾ𝑎𝑐𝑔𝑡ሿ𝑘

EXHAUSTIVE SEARCH DEPENDENCE

Naive method Enumerate all the combinations and find the max

likelihood combination. Example: length 5

1,2,3,4,5 (1,2,3),4,5 (1,2)3,(4,5) (1,2,4,5)3 (1,2,3,4,5) ….

EXHAUSTIVE SEARCH DEPENDENCE

improved method: Enumerate only single dependence group

If D1 and D2 are two independent groups Then D1, D2 can be used to compute

D1,D2 In fact, greedy search

Example: sorted combination (log likelihood) (1,2),3,4,5: -32 (1,2,3),4,5:-44 1,2,3,(4,5):-50 … 1,2,3,4,5:-100

The best (1,2),3,(4,5)

RESULT

Run MEME, Cisfinder, Amadeus, ChIPMunk, HMS, Trawler, Weeder, JPomoda on 15 ES ChIPseq datasets

Using one half of ChIPseq peaks to learn de novo PWM, and the other half to validate their performances.

RESULT

MEME DP_MEME Weeder DP_Weeder Cisfinder DP_Cisfinder Amadeus DP_Amadeus HMS DP_HMS trawler DP_trawler ChIPMunk DP_ChIPMunk Jpomoda DP_Jpomoda

tcfcp2I1 0.9212 0.9375 0.8911 0.9544 0.9328 0.9644 0.8615 0.8752 0.9707 0.9673 NA NA 0.9703 0.9702 0.9710 NA

klf4 0.8625 0.8596 0.8445 0.8569 0.8487 0.8601 0.8240 0.8389 0.8612 0.8561 0.6310 0.6538 0.8637 0.8592 0.8360 0.8369

suz12 0.6434 0.6438 0.5852 0.5695 0.5760 0.5838 0.5912 0.5919 0.5920 0.5959 NA NA NA NA 0.5963 0.6005

zfx 0.7586 0.7548 0.7717 0.7432 0.7406 0.7433 0.6974 0.7089 0.6166 0.6096 0.7606 0.7624 0.7562 0.7672 0.7522 0.7531

stat3 0.7137 0.7229 0.6989 0.7200 0.7216 0.7323 0.7159 0.7041 0.7035 0.7116 0.6898 0.7090 0.7243 0.7332 0.7455 0.7424

nmyc 0.7785 0.7803 0.7425 0.7406 0.7494 0.7520 0.7425 0.7455 0.7145 0.7358 NA NA 0.7547 0.7728 0.7640 0.7602

esrrbredo 0.9099 0.9052 0.9076 0.9144 0.8994 0.9051 0.8874 0.8820 0.8807 0.8967 0.8769 0.8876 NA NA 0.8729 0.8713

cmyc 0.7681 0.7668 0.7550 0.7617 0.7746 0.7728 0.7631 0.7594 0.6855 0.6984 0.7472 0.7675 NA NA 0.7801 0.7807

e2f1 0.6110 0.6185 0.5714 0.5803 0.5729 0.6039 0.5818 0.5875 0.5884 0.5900 0.5629 0.5767 0.6420 0.6464 0.6208 0.6204

nanog 0.6690 0.6813 0.6649 0.6850 0.6635 0.6835 0.6074 0.6171 0.6722 0.6795 NA NA 0.5635 0.5554 0.6964 0.6997

oct4 0.6673 0.6827 0.6646 0.6816 0.6460 0.6784 0.6293 0.6470 0.4790 0.4780 NA NA 0.7194 0.7136 0.6880 0.6891

sox2 0.8449 0.8837 0.8151 0.8514 0.8145 0.8615 0.7369 0.7434 0.5758 0.5823 0.7881 0.8506 0.8323 0.8558 0.8185 0.8427

smad1 0.5848 0.5847 0.5847 0.6042 0.5765 0.5765 0.5767 0.5767 0.5781 0.5718 0.5484 0.5504 0.6048 0.5957 0.6328 0.6328

ctcf 0.9809 0.9854 0.9708 0.9846 0.9648 0.9855 0.9474 0.9680 0.9790 0.9862 NA NA 0.9819 0.9818 0.9804 0.9835

p300 0.6198 0.6062 0.5355 0.5224 NA NA NA NA 0.5749 0.5883 NA NA 0.5831 0.5898 0.5709 NA

ADJACENT DEPENDENCY

MEME CTCF motif 1-2-3,10-11 AUC Result:

MEME:0.9809 Dependence: 0.9854

LARGE DEPENDENCY GROUP

MEME SOX2 motif 1-2-3-4-5-7,14-15 AUC Result:

MEME:0.845 Dependence:0.884

LONG DEPENDENCY

MEME NMYC motif 10-21,11-12 AUC Result:

MEME:0.7785 Dependence: 0.7803

NEW SERVERS CONFIGURATION

MODEL & PRICE

Hostname: genome3U server2X Intel Xeon X5680 Processor(6-core each) 144GB RAM16X2TB SAS　 Disks2X1G network interfacesPrice:20kSGD

Hostname: biogpu1U server2X Intel Xeon X5680 Processor(6-core each) 2XM2050 GPU48GB RAM3X2TB SATA2 Disks2X1G network interfacesPrice:18k SGD

FILE SYSTEM

genome: RAID-6, 28TB , Centos5.5 Home:23TB

biogpu: RAID-5, 4TB , YellowDog linux (Centos5.4) Home: 3TB

SERVER SOFTWARE

NIS: using the same account for 2 servers NFS:

Home directory : genome server Public_html: biogpu server Share software: /cluster/biogpu/programs/bin/

Apache: biogpu server Mysql: genome server

CURRENT PROBLEMS

Filesystem Size Used Avail Use% Mounted on/dev/mapper/VolGroup00-LogVol02 393G 4.7G 368G 2% //dev/mapper/VolGroup00-LogVol00 2.0T 199M 1.9T 1% /tmp/dev/sdb2 23T 23T 439G 99% /home/dev/sda1 920M 47M 826M 6% /boottmpfs 71G 0 71G 0% /dev/shm

I/O killer

TO DO

Swap backup Connect to Tembusu Install SGE

GPU COMPUTING

FERMI M2050 Fermi M2050

Peak double precision floating point performance

515 Gigaflops

Peak single precision floating point performance

1030 Gigaflops

CUDA cores 448

Memory size (GDDR5) 3 GigaBytes

Memory bandwidth *(ECC off)

144 GBytes/sec

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CODE EXAMPLE TO ADD TWO ARRAYS CUDA C Program

__global__ void addMatrixG( float *a, float *b, float *c, int N ) int i = blockIdx.x * blockDim.x + threadIdx.x; int j = blockIdx.y * blockDim.y + threadIdx.y; int index = i + j * N; if ( i < N && j < N ) c[index] = a[index] + b[index];

void main() ...... dim3 dimBlk( 16, 16 ); dim3 dimGrd( N/dimBlk.x, N/dimBlk.y ); addMatrixG<<<dimGrd, dimBlk>>>( a, b, c, N );

Device code

Host code

A CUDA kernel

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CUDA MEMORY MODEL Each thread can

R/W per-thread registers R/W per-thread local

memory R/W per-block shared

memory R/W per-grid global

memory RO per-grid constant

memory RO per-grid texture

memory

Host can R/W global, constant and texture memory

Host

CUDA MEMORY HIERARCHY

The CUDA platform has three primary memory typesLocal Memory – per thread memory for automatic variables and register spilling.

Shared Memory – per block low-latency memory to allow for intra-block data sharing and synchronization. Threads can safely share data through this memory and can perform barrier synchronization through _ _syncthreads()

Global Memory – device level memory that may be shared between blocks or grids

MOVING DATA…CUDA allows us to copy data from one memory type to another.

This includes dereferencing pointers, even in the host’s memory (main system RAM)

To facilitate this data movement CUDA provides cudaMemcpy()

31

CUDA EXAMPLE 1 – VECTOR ADDITION (1)

// Device code__global__ void VecAdd( float *A, float *B, float *C ) int i = blockIdx.x * blockDim.x + threadIdx.x; if ( i < N ) C[i] = A[i] + B[i];

// Host codeint main() // Allocate vectors in device memory size_t size = N * sizeof(float); float *d_A; cudaMalloc( (void**)&d_A, size ); float *d_B; cudaMalloc( (void**)&d_B, size ); float *d_C; cudaMalloc( (void**)&d_C, size );

32

CUDA EXAMPLE 1 – VECTOR ADDITION (2) // Copy vectors from host memory to device memory // h_A and h_B are input vectors stored in host memory cudaMemcpy( d_A, h_A, size, cudaMemcpyHostToDevice ); cudaMemcpy( d_B, h_B, size, cudaMemcpyHostToDevice ); // Invoke kernel int threadsPerBlock = 256; int blocksPerGrid = (N + threadsPerBlock – 1) / threadsPerBlock; VecAdd<<<blocksPerGrid, threadsPerBlock>>>( d_A, d_B,

d_C ); // Copy result from device memory to host memory // h_C contains the result in host memory cudaMemcpy( h_C, d_C, size, cudaMemcpyDeviceToHost ); // Free device memory cudaFree(d_A); cudaFree(d_B); cudaFree(d_C);

OPTIMIZATION

Minimize the diverse path (if,else …) Collapsed access global memory Scattering to gathering Use Share memory as much as possible

34

Compiling codeLinux

Command line. CUDA provides nvcc (a NVIDIA “compiler-driver”. Use instead of gcc

nvcc –O3 –o <exe> <input> -I/usr/local/cuda/include –L/usr/local/cuda/lib –lcudart

Separates compiled code for CPU and for GPU and compiles code. Need regular C compiler installed for CPU.Make files also provided.

Windows

NVIDIA suggests using Microsoft Visual Studio

CUDA TOO HARD?

Use others software with cuda acceleration Use wrapper library

CUDA ACCELERATED SOFTWARE

cuBlas, cudaLAPACK CudaR, CudaPy Cuda Bioinformatics Softwares:

Molecular Dynamics & Quantum Chemistry • ACE MD • AMBER • BigDFT (ABINIT) (news) • GROMACS • HOOMD • LAMMPS • NAMD • TeraChem (Quantum Chemistry) • VMD

Bio Informatics • CUDA-BLASTP • CUDA-EC • CUDA-MEME • CUDASW++ (Smith-Waterman) • DNADist • GPU Blast • GPU-HMMER • HEX Protein Docking • Jacket (MATLAB Plugin) • MUMmerGPU • MUMmerGPU++

THRUST

Searching Binary Search

Vectorized SearchesCopying

Gathering Scattering

Reductions Counting Comparisons Extrema Transformed Reductions Logical Predicates

• ReorderingPartitioningStream Compaction

• Prefix SumsSegmented Prefix SumsTransformed Prefix Sums

• Set Operations• Sorting• Transformations

FillingModifyingReplacing

EXAMPLE 1

#include <thrust/count.h> #include <thrust/device_vector.h> ... // put three 1s in a device_vector thrust::device_vector<int> vec(5,0); vec[1] = 1; vec[3] = 1; vec[4] = 1; // count the 1s int result = thrust::count(vec.begin(), vec.end(), 1); // result is three

EXAMPLE 2 #include <thrust/transform_reduce.h>

#include <thrust/functional.h>#include <thrust/device_vector.h>#include <thrust/host_vector.h>#include <cmath>

// square<T> computes the square of a number f(x) -> x*xtemplate <typename T>struct square __host__ __device__ T operator()(const T& x) const return x * x; ;

int main(void) // initialize host array float x[4] = 1.0, 2.0, 3.0, 4.0;

// transfer to device thrust::device_vector<float> d_x(x, x + 4);

// setup arguments

square<float> unary_op; thrust::plus<float> binary_op; float init = 0;

// compute norm

float norm = std::sqrt( thrust::transform_reduce(d_x.begin(), d_x.end(), unary_op, init, binary_op) );

std::cout << norm << std::endl;

return 0;

40

REFERENCES NVIDIA CUDA Programming Guide, Version 2.3

http://developer.download.nvidia.com/compute/cuda/2_3/toolkit/docs/NVIDIA_CUDA_Programming_Guide_2.3.pdf

NVIDIA CUDA C Programming Best Practices Guide, Version 2.3 http://

developer.download.nvidia.com/compute/cuda/2_3/toolkit/docs/NVIDIA_CUDA_BestPracticesGuide_2.3.pdf

http://code.google.com/p/thrust/wiki/Documentation