Problem 1 [A Few Random Steps} Consider four molecules, all starting at the origin, which undergo a 1d random walk, each with a 50% chance of going left or right along the x-axis at each time step. A. After 1 time Step, what is the probability that 2 particles have gone left and 2 have gone right? Give your answer as a decimal number between G and 1 to 3 decimal places. B. After 2 time steps. what is the probability that all of the particles are at the origin? Give your answer as a decimal number between G and 1 to 3 decimal places. C. After 2 time steps. what is the probability that none of the particles are at the origin? Give your answer as a decimal number between G and 1 to 3 decimal places. D. After taro time steps. what is the probability that one or more particles are not at the origin? Give your answer as a decimal number bets-seen D and 1 to 3 decimal places. Problem 2 (10 random walk) Consider a large number of molecules starting at x = 0 undergoing a 10 random walk (i.e. diffusing in 10). Each time step these molecules move a distance of 5 um (either left or right), and the molecules move with a speed of 800 um/s. A. What is the diffusion coefficient for these molecules? Give your answer in units of Hm /s. B. After these molecules have undergone a random walk (i.e. diffused) for 50 seconds, what is the average x-position of a molecule in this collection? Give your answer in units of um. C. After these molecules have undergone a random walk (i.e. diffused) for 50 seconds, what is the rms-average distance from the starting location of a molecule in this collection? Give your answer in units of um. D. After 50 seconds, what is the farthest one of these molecules could possibly have traveled from the starting location? Give your answer in units of um.E. If initially there are 1030 molecules diffusing, how many have gone the maximum possible distance (most likely)?Problem 3 (Sequencing DNA) Suppose we have access to the four nucleotides which compose DNA: A, T, C and G. Placed into a sequence with one another, they form a code which can be read by the cell to provide a blueprint for the creation of proteins. When forming a sequence of three nucleotides, the sequences TAA, TAG, and TGA are codes to tell the cell to stop reading the DNA. Assume each sequence of nucleotides is equally likely to be formed. A. How many possible ways are there to form a sequence of three nucleotides? B. Of the possible ways to form a sequence of three nucleotides, what is the probability to form a stop sequence? C. What is the probability that a sequence that is not a stop sequence is formed?D. For a three-nucleotide sequence, G is engineered to be in the third place. For the remaining two nucleotides in the sequence, C cannot be in the second place (for example CTG is allowed but not TCG). How many sequences can be formed? E. Of the number of sequences found in Part D, what is the probability that a stop sequence is formed?Problem 5 (Why you should stir) Calculate the time it takes for glucose, initially concentrated at a point, to spread out so that the rms-average distance traveled by the glucose molecules is 1 cm (D = 6.7 x 10-10 m/s?). Express your answer in seconds. Problem 6 (Diffusion by a protein in water) Estimate the diffusion constant of a spherical protein of radius 5.0 nm moving through water at room temperature (300 K). The viscosity of water is 1 mPa s. Express your answer in um*/s.Problem 9 (Oxygen diffusion in a bacterium] The size of spherical aerobic bacteria is limited by the rate at which oxygen diffuses through water. A bacterium with a radius greater than about 10 um is not able to obtain enough oxygen from the surrounding water to sustain itself. Given that the diffusion constant of oxygen in water is 8 x 10-10 m's 1. How long does it take for a collection of oxygen molecules to diffuse a rms distance of 10 um? Express your answer in ms. (Hint: movement is in 3-dimensions.)Problem 10 (ATP diffusion) The diffusion constant of ATP is 3 x 10-10 m's 1. How long would it take for an ensemble of ATP molecules to diffuse a rms distance equal to the diameter of an average cell (diameter -20 um)? Express your answer in ms. (Hint: movement is in 3-dimension.)Problem 13 (Random walks) A marker in a game is constrained to move along a one-dimensional grid. It begins at 0 and moves according to a coin flip: left for heads, right for tails. After 3 flips it will be either 1 step away from its starting point or 3 steps away. +2 +3 A. How many different sequences of 3 coin flips will lead to the coin ending at -1?Problem 14 (Amoeba random walking) In this problem you will work out a simple model for an amoeba's quest for dinner. This amoeba lives in a G two-dimensional world, on a surface conveniently divided into grids. For our purposes, the surface is infinite in extent, and everywhere the same. Part of the infinite plane is shown at right. In a time interval At the amoeba can move exactly one grid to the right (R), one grid to the left (L), one grid up (U), one grid down (D), or it might stay (S) B where it is. You may assume the amoeba's decisions are completely random, so that each possibility is O equally likely. We begin the observing the amoeba at time t = 0, with the amoeba at point X. A. After a time of only one At, how many different ways could the amoeba have ended up at each the following points (i.e. how many distinct paths with a length of 1 step end at the given point)? A ii. B iii. C iv. X V. How many total paths are associated with one unit of time At?B. After two units of time, (so = 2At ), how many different ways could the amoeba have ended up at each of the following points? i. A ii. B iii. C iv. X V. How many total paths are associated with time t = 2At? vi. What is the probability P that the amoeba will be found at the point Cat time t = 2At?Problem 15 (Diffusion across a membrane) Consider potassium ions crossing a biological membrane 10 nm thick. The diffusion coefficient for potassium in the membrane is 1.0 x 10-16 m /s. (a) What is the net number of potassium ions per second that will move across an area 100 nm x 100 nm if the concentration difference across the membrane is maintained at 0.50 mol/liter? Give your answer as a positive number in units of numbers of ions.D. Assuming you had 100 markers, and performed 3 flips for each marker moving them according to the above rules, which of the following would be true about the distribution of markers? Select all that apply a. All the markers would be at the starting position b. More markers would be at the starting position than any other location C. The most markers would be at +1, and there would be no markers at the starting location. d. There would be exactly equal numbers of markers to the left and to the right of the starting position e. The average distance from the origin of the markers would be a number between 1 and 3