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Cells in a multicellular organism (e.g., figure below) oftensignal to each other by having specialized cells that emitsignaling molecules. Those signaling molecule...

Question

Cells in a multicellular organism (e.g., figure below) oftensignal to each other by having specialized cells that emitsignaling molecules. Those signaling molecules diffusethrough the fluid medium until they are detected by other cells,which triggers some sort of biochemical or biomechanical response.Consider two cells that communicate by diffusing a chemical thathas a diffusion constant, D = 250 μ m 2 /s. The cells are a few μ m in diameter, whichis much smaller than the distance between the

Cells in a multicellular organism (e.g., figure below) often signal to each other by having specialized cells that emit signaling molecules. Those signaling molecules diffuse through the fluid medium until they are detected by other cells, which triggers some sort of biochemical or biomechanical response. Consider two cells that communicate by diffusing a chemical that has a diffusion constant, D = 250 μ m 2 /s. The cells are a few μ m in diameter, which is much smaller than the distance between them. For the purposes of this assignment, treat the situation as if it were simple diffusion in one dimension, 〈 r 2 〉 = 2 D t, where r is the distance moved by the signaling molecule in a time t. Any given molecule can move a random distance, but the average distance moved by many particles is well defined. 1. Consider two of these cells separated by a distance of 1 mm ( 10 3 μ m ), as shown below. On average, how long would it take for signaling molecules to diffuse between the two cells? 2. Let's see what happens if we have 10 times the original distance (r = 10 m m) and only two cells? On average, how long will it take signaling molecules to diffuse from one cell to the other? (Give your answer as a factor of the time it takes compared to the time from the preceding question, e.g., 10X, 100X, etc.) 1. If the ATP hydrolysis rate for a certain motor is 50 ATP molecules per second, at what rate is chemical energy released? Answers are given in Joules per second, J s. (Recall Avogadro's number: 6.022 ⋅ 10 23)



Answers

a. Many biochemical reactions that occur in cells require relatively high concentrations of potassium ion (K $^{+}$ ). The concentration of $\mathrm{K}^{+}$ in muscle cells is about $0.15 \mathrm{M} .$ The concentration of $\mathrm{K}^{+}$ in blood plasma is about $0.0050 M .$ The high internal concentration in cells is maintained by pumping $\mathrm{K}^{+}$ from the plasma. How much work must be done to transport 1.0 mole of $\mathrm{K}^{+}$ from the blood to the inside of a muscle cell at $37^{\circ} \mathrm{C}$ (normal body temperature)? b. When 1.0 mole of $K^{+}$ is transferred from blood to the cells, do any other ions have to be transported? Why or why not? c. Cells use the hydrolysis of adenosine triphosphate, abbreviated ATP, as a source of energy. Symbolically, this reaction can be represented as $$\operatorname{ATP}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{ADP}(a q)+\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q)$$ where ABP represents adenosine diphosphate. For this reaction at $37^{\circ} \mathrm{C}, K=1.7 \times 10^{5} .$ How many moles of ATP must be hydrolyzed to provide the energy for the transport of 1.0 mole of $\mathrm{K}^{+}$ ? Assume standard conditions for the ATP hydrolysis reaction.

This question has five parts to it. This is the first part. We've been given some information and we're being asked to solve for how big would the cell appear? Uh huh. And so and back in 2000 times helping what it appear. Magnification is the size of the image, divided by the size of the object. So we have been given both the magnification and the size. So what you do is take the total size of image to find that you would take 10,000 times 50 to be 500,005 kilometers heartbeat. What's the number of act? Yeah, And within a mile site no attacked diameter of 3.6 for the radio. One point a volume is for third Pirates fire, third power. So what? You have 1.34 3rd times 3.1 more touch five point game for you. That's what. My name in the system. So when you get a 13.3, how much do you get the volume putting 50 micrometers. Yeah, your appearance with the cell being 50 micrometers, the radius is 25 micrometers 25. Cute, of course, is 15,625 and then times four thirds times pi, you get 63,781 0.25 So then the equation comes to 13.39416 size of one acting volume, times x You don't know how many is gonna be There equals volume of a cell 63.7. I'm sorry. 63,781 0.25 Divide both sides by 13.39416 x which is the number of active? Yes, Part two c Part C's where we have Okay, my country. And here is the final answer for part being 4761 0.87 Yeah, which is a 1.5. We are to assume a spear shape to it. So if the diameter is 1.5 on the radius is 0.75 take the 0.75 cube multiple applied by 3.14 and then also by 1.3. What you get is 1.7 2209375 This is the volume of one mitochondria. Now compare that with the volume of a single cell, which is the typical cell. 63,781 10.25 Do that comparison Onda Setting up your equation 63,781 0.25 equals 1.72 other numbers. Times X. Then you'll take X equals and you'll have 63,000 divided by 1.72 And what we get is 37,037 0.4 as the number of mitochondria if it's within that stuff. Yes, hurt thing is when you have the concentration of consciousness, one million more leaders in the question is asking how many molecules first thing is to understand. The other address number states that for every one bowl you have to explode on two types 10 to 20 1300 molecules. So the big thing is Thio brave Billy Bowl to a comparative mole compared for balls volumes. So what we're gonna do is take the one million mole and understand that this is over. Leaders understand the one million mole and then divide by 1000 million holes for everyone home and then take that number now that it's in bowls and converted into molecules by multiplied by 6.2 times. Attention car. So there is one of a couple of ways you could get a change. These two types Texas, with their power divided 5000 in the 1000 essentially as one times 10 to the third power. So 23 minus 30 20. So it's the same numbers because on the tops exporters here now it's times 10 to the 20th power. Alternatively, you can also write out all of the heroes and all the place that values have. A doctor has never We're not going to do that today. Car eyes relative Thio it essentially Get on this says you got 20 micrometers. I'm sorry. Uh huh. Close thing. This is in concentration. So basically, my God, is that and then you also have for the cellular concentration for the coast being one billion gold for later. So then you make a comparison, and then you're going to see well, how many molecules per Hexham, Kanye's molecules. So the way to do this is to take first of all, the 20 micro mole per leader and, uh, do that for later. In just a minute But the thing is talking about mole aspect. So 20 micro mole times one million mole divided by 1000 Micro moles times one mole This is again mold for later, divided by one million hole. And so, essentially, what you have is 20 divided by a million. And what you end up with his 0.2 or two times 10 to the negative Fifth. This would be in terms of molds and then taking that and converting it to molecules. You're gonna have point again. 00002 types, 6.2 times 10 to 23rd molecules. Since its Afghan dress number is 6.2 times 10 to 23rd molecules wherever one bowl would you do that? You have essentially that. You've got two times 6.2. That's gonna be 12.4. And then you have, uh, the expel its sits there being multiplied, they're gonna add together and negative five plus 23. So the exponent you'll have for 12, 24 times 10. 18. Then you've got to put that back into scientific notation. So will be 1.204 times 10 to the 19 power. So then the next step is comparing the amount of molecules that we found in G, which was the glucose being in the one million mole that you had 6.2 times 30th power number of molecules compared to the hex of hiding molecules. So 6.2 times 10 to 20 power divided by 1.204 times 10 to 19. Power that again. You can do this one of a couple ways. One way. It just simply take 6.2 divided by 1.204 and what you're gonna end up with, Yeah, six divided by 1.2. It's like, uh, 0.5 point because the decimal places it's gonna be a 0.0.5 and then you have the 20th power and then the advisor in the 19th powers Decision Adviser, they're going to subtract the SP and so 0.5 times 10 to the first power because 20 months, 19 1, and it ends up being 10.5 times 10 to the first power or that is able to find the alternative

For this question. We're trying to understand how C T cells were to present antigen from different strains of, say, a flu virus. So in this experiment, they're using two different flu virus strains. They have a 1968 flu virus and a 1934 flu virus. And what they are doing is they are making proteins from the genetic sequences of each of these, and they're finding out that the 1968 gives a response by the T cells, whereas the 1934 strain has no T cell response or very little. And this is measured by the number of cells that are being Liszt when they contain the proteins from each of these flu strains, so they give us a graph. To represent this, they have the number of cells, or the percent of cells, life on the Y axis. And they're presenting us with multiple bars measuring the number of cells being laced so they have a bar for the 1968 virus, a bar for the 1934 virus, and you can see there's a significant difference between the number of cells BIST for each of these strains from there they are measuring the number of cells that lies when they contain a certain amino acid sequence. So the first section is for cells containing amino acids from the 1968 stream. But as the second sequence of bars are amino acid sequences from the 1934 strain. So as you can see in the 1934 strain, you have two different sequences. You have a 3 65 to 3 80 and you, of course, have a 3 69 to 3 82. And both of these have very little cell isis among all the cells, whereas in the 1968 we have three different bars, the most significant ones being the one showing many cells being Liszt, which is the 3 65 23 80 and the amino acid numbering sequence. And you also have 3 69 23 82. So between these two amino acid sequences, the difference between each of these is causing this significant change in the number of cells being recognized by your T cells and being Liszt. So the main difference in amino acid sequence between these two it's going to be those first few starting amino acids. So here, for the first part of question a the amino acid sequence responsible for this T cell recognition are going to be the amino acids 365 to 369. We know that 369 to 380 is not causing this response. Or else we would see the bar in this second part for the 1968. Also induce that large number of style license. So here for part two, a question A. We're also being asked why these other viral proteins do not cause this significant sell like Isis. That's because of the recognition of the anti Jinbei your T cells. T cells are not all powerful, and they're not going to recognize and induce apoptosis in all kinds of cells with viral antigens. It just so happens that they can recognize some sequences, and by chance, they happen to recognize the 365 2, 369 amino acid sequence. If they recognized all amino acid sequences, they risk the chance of endangering our own cells, which, of course, have sequences of amino acids. And this limited recognition kind of helps that specific response of T cells and prevent them from overreacting. But I say we have part B. It's asking us about how the MHC type one molecule works and how it's applied to this experiment. So if you remember how MHC molecule works specifically MHC one, this occurs in all living cells. And when they have, say, a viral strand inside their genomic structure, they're going to create viral proteins to have these be viral proteins. The cell is going to recognize that these viral proteins are not their own. They're going to connect them to an MHC molecule. And when it's complex with the viral protein, it's going to move it to the surface of the cell, where it can present this auntie gin to your professional antigen, presenting cells such as your B cells or macrophages, and that will allow for your cell mediated response and the like isis of the cells. So, in order for this to work for this experiment, they're measuring the number and percent of cells being laced. So the best way to go about this is there going to use a 96 well plate, which is basically just 96 specimen holders where they're going to have the infected cell on the inside containing these viral proteins. They're going to line these 96 well plates with the cells they're infecting, and they're going to infect them with different amino acid sequences. So this would be like an amino acid strands of 3 65 23 80. Or they might do the 3 69 to 3 82. And they're going to do this for each the 1968 strain as well as the 1934. And to make sure they can measure the number of cells Liszt. They're going to need a probe or some sort of marker that they can check inside of the well. So what they're going to do is to each of these amino acids, the viral protein or the viral genetic sequence they're going to infect each of these cells with is going to include what we call a probe or some sort of marker. So this could be a marker that has some sort of fluorescence. It could be some sort of radioactive marker, something they can use to detect and measure the amount in the super needn't or the remaining fluid in the 96 well plate. Basically, it's going to be included inside the cell, and if the cell lice is, it's going to be released into the fluid of the 96 well, plate, whereas if it does not light, it will still remain inside the cell, where it won't be measured in the fluid. So what they're going to do is there going to centrifuge each other? 96 well, plates, and they will get a button of live cells on the bottom, and they will get a number of the probe for the marker for the concentration inside of the fluid or the super name. So they're they're able to compare the number of cells they initially put in and compare that two. The amount of probe in the Super Natan, where this would be the inverse of it, will be the probe divided by the cells, and that would give them the percentage of cells that were laced. So here, for part B, they're going to introduce the virus and some sort of probe into the cells, and from there they're going to compare the amount of probe and the Super Matan to the number of cells they started with, and that's going to give them the percentage of cells that were Liszt in the solution. This way, they'll be able to reliably compare the number of live cells and not include that in the number of life cells they originally started out with.

Yeah. All right. Let's consider a scenario in which we have to variation isotopes. Two flavors hydrogen. The first one will just call hydrogen. Only one called. I might be butchering this pronunciation here, but do terry. Um Mhm. Okay. So if both of the temperatures are equal, which can we expect to have a greater kinetic energy? So, you might recall a kinetic energy can be evaluated Music 3/2 Ultimate constant two times temperature quote kinetic energy. This relationship was it can be determined experimentally because temperature is directly proportional to the speed of the particles in the gas. So I do not expect that either of them have greater kinetic energy. I'm moving on. Which diffuses faster at a given temperature. Okay, So we can also solve for kinetic energy using this expression right here. Now, we know that the kinetic energy is equal because the temperature is equal. We would expect this expression to betrayal. Uh huh. Okay. Now say we Okay, say for hydrogen we have a decrease in mass or it's decreased relative to deuterium. However, if this were to happen and for this relationship to hold for any given set of atoms particles hears me. I expect the velocity hydrogen also need to increase break even with the kinetic energy vegetarian. So I expect use the astra. It's a strange looking at mhm. Yeah. Sloppy cursive much better. All right. So, we're giving a problem in which given the weight of hydrogen and atomic mass of united. So, that's just one atomic mass unit, there's dietary um that's too topic fast minutes. Find the ratio of the diffusion rate. It doesn't specifically specify uh what that ratio is supposed to look like, whether or not it's the hydrogen, the fusion rate over the dietary infusion rate or vice versa. However, let's just solve it on the assumption that it's supposed to be the ratio of deuterium to hydrogen so established this just a few months ago. So if he said oh Mm subscript age over 1/2 and subscript E sub script D squared over the subsequent age squared, cancelling out that pesky sign and cleaning it up just a little bit were locked with the following expression. Okay, so you can find the ratio of the velocities which can tip us off to the velocity uh infusion rate. Um because random walk is more or less equivalent regardless of the speed that you're going. Therefore the diffusion rate is directly proportional to the velocity. Therefore this ratio is valid. Are given the information we dissolve the ratio of em subscript age to m subscript E. So let's go ahead and solve this real quick and subsequent page one, I am the subscript. Let's do atomic mass units. Find it square root All for it. We get 707. So that suggests to us the velocity. Mhm, the velocity to age his last than want. In other words, the subscript E. Is less than the subject age. Now, you should see this is perfectly consistent with what we said earlier, hydrogen fuses faster. Smaller objects are faster. No surprises there. So we know that hydrogen will diffuse faster because it's translational velocity is higher. I hope that helped you.

Problem Number 77 in this problem, it is given that in a hydrogen atom, an electron is moving in circular orbit at Honda rode on in the hydrogen atom. And we have to find the velocity V off the moving electron, the radius off the electron, but is given us our our equals 5.29 in tow. 10 to the power, minus 11 meters. This is the radius off Norbit. We know that must off electron. Mm. He's given us 9.1 into 10 to the power minus 31 kitty. And the charge on a proton and electron. You is he called. Oh, 1.6 into 10 to the power minus 19. Cool. Abs. You know that between on electron and proton, there exists on electrostatic force. This electro is static. Force will provide the records centripetal force for the electron toe move in the circular path around a proton so we can write electrostatic force. Is it Gordo centripetal force thus? Okay. Que square upon rd squared is equal to myself. Electron in tow. Be square. My canceling our from both sides figure. Okay, you square. Bye. Oh, God. Is it cool toe M V you square. There. Okay, is columns constant and ritual is nine into turned to the power nine. You Didn't we do this square, Markkula, Me Squire. Now substituting the values we get nine in tow. Tend to the power nine in tow. 116 into 10 to the power minus 19. Square upon 5.29 into 10 to the power minus 11 is equal to mass off electron, which is 9.1. You do tend to the power minus 31 we square. This implies that we square equal to 478 0.6 into 10 to the power minus three into 10 to the power minus 38 into 10 to the power nine. Bye. 10 to the power minus 11 to tend to the power minus 31. On solving this expression, we will get the Villa Street off the moving electron as 2.1 it into 10 to the power six meter, four seconds. This is the required velocity off the electron moving in the circle.


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